Beginning in the middle is absolutely fine - in fact, how else could it be? We should come aboard the process 'in the middle, as onto a moving train' (I read this somewhere - it is either Gilles Deleuze or someone like him); how else can we seize hold of the story in its authentic form?
Personally, I can't imagine any way of starting to write a piece except to jump in at the middle! Lexi's answer - to write, and then to rewrite - is an excellent method. Where is the shame in improving on one's first random scrawls? We cannot expect, like Shakespeare, never to revise our work (and even his habit of never 'blotting' or amending his first drafts had its detractors - as his contemporary Ben Johnson put it, 'Would that he had blotted a thousand!').
There is one phrase that worries me in the question: 'particular scenes or moments which I think will be crucial'. Crucial to what? Do you already have a plot mapped out - Character A will do this, then do that, then the other thing? If so, I strongly recommend you reconsider this method. When we write, we learn; we feel our way into the world of the story. The inner logic out of which the narrative arises cannot be written out in our heads ahead of time - even if it were, what would be the point of writing the story? One could just as well tell the skeletal plot in a few words and then forget about.
The story will necessarily be inauthentic (at least, this is my experience) if we already know what is going to happen. We are not learning. The analytical side of one's brain has taken over way too early; the other side of our brain is the one we need when are writing a first draft - the one that muddles through, the bricoleur, the synesthesiac. It is this side that learns what the story is. The analytical side can come in later when the story is there already and needs to be improved.
Therefore: blindfold yourself, and leap! And begin in the middle - though for all you know, the middle may turn out to be the beginning, or the end, or no part of the story at all - a MacGuffin you will later discard.