I'm writing a screenplay with a scene were characters interrupt each other frequently. I think including the lines a character would've said if they weren't interrupted could help the actors and directors. Is there a standard way of doing this?
Write the lines in full.
Indicate in the stage directions that they speak over each other and interrupt.
As a scriptwriter: do not attempt to micromanage the performances of actors who do not yet exist! This is not your job, it is the job of the director and actors themselves.
If you're writing for self-enjoyment with no intention of having your script produced, then it doesn't matter how you choose to format the dialog. Write it as you like.
But if you're writing a professional screenplay, you need to understand the script will not be treated with any sanctity or precision – if it is, it's to the detriment of human professionals who understand their part of the job including the director, the actors, the editor, and any Producer decisions that need to be made on set.
The Thing from Another Planet
I suggest you find a film where the actors speak in the manner you want, and try to find the original screenplay to compare how it was written. An easy-to-find example is The Thing from Another World (1951).
The film benefits from fast pacing and a pattern of dialog where all the characters speak over each other. The dialog that ended up onscreen is not the same as written. It follows closely, and the script is dense with short lines that feel spontaneous and conversational (there are almost no long expository speeches), but many lines were improvised on set that adjust the pacing and temper the characters. Some dialog that feels more confrontational (argumentative) in the script becomes playful and slightly irreverent in the final film – they way professionals actually speak to each other when they are familiar and tired, something between banter and sarcasm.
But, this one line makes the whole scene...
There's a writing adage: "A scene is not a line; a script is not a scene." The meaning is that your scene cannot hang on one line of dialog (and the longer script cannot hang on one scene).
If you have a specific line of dialog where it's plot-crucial that it gets interrupted (because a character will misunderstand or some info is misinterpreted), write it as interrupted dialog. The context should carry the meaning. If the lines require explanation you have a problem.
Screenplays are pre-production documents that do not survive the production un-changed. Remember that everyone involved in realizing the actual production will have ample opportunity to study and discuss the script. Scenes will be blocked and paced by the director, individual lines will be shot out-of-order according to production needs. Entire scenes will be cut.
More context is added through the actual production –– the acting and framing, as well as innate visual and temporal cues that have no business in a screenplay. The original writer is rarely on set during production. The professionals creating the final film do not need to be constrained by a 'dead' creative voice that made qualitative decisions months or years before the production started.
The standard way is to put it on the screen. They can say it later in dialogue, perhaps. Another character can ask them, later.
If you cannot fit that in, leave it out.
The number one rule in screenplays is put it on the screen or leave it out.
The audience only sees what is on the screen, nothing else. They can see if your actor is frustrated, angry, hateful, or despondent, amused, whatever. You can describe that.
The director and actors are not going to infer from whatever you wanted that character to say how to act the scene. Just tell them how to act the scene, what emotions the character is feeling. That is all they want to know, what can be seen/heard on the screen.
Failing to follow this rule increases the chances of rejection dramatically. You must write from the POV of the audience, if the audience cannot see or hear it, if an actor cannot portray it, leave it out. And be specific about what they are supposed to portray -- resigned frustration, angry frustration, resentment, despair? What is it you want the audience to see?
That is all the director and actor wish to know. They will read the entire script, spoilers and all, before they ever act the first line. They will read it multiple times.
That is something you can count on, if you wanted the character to say something about a future plot point, the actors and director will know what it is when the interruption is acted.