I actually asked a similar question to this in a different SE site, and I think that GRRM actually provides enough evidence to prove that he is an incredibly accomplished swordsman.
However, you are correct, that he does the majority of building up Jaime's legendary skill with telling rather than showing, but after researching it I realized that this was entirely intentional.
Yes, he is an accomplished swordsman, and we see it on several occasions, and others who have seen him fight can vouch for that. However, he underwent a very unique set of circumstances to build up the renown around the world that he is the greatest swordsman alive.
If you read the linked question, I've provided all the reasons for how his natural skill, luck, and also politics contributed to his legend. And this legend surrounding him causes him to become one of the most arrogant and cocky characters in the book, assuming that he can do as he pleases and get away with it because no one can stop him.
This was all to build up to the pivotal moment when he lost his hand. As the author loves to kill off the main characters, this was simply another one of them. He killed the greatest swordsman alive.
Throughout the rest of the story, Jaime goes through a massive transformation in character, as he reevaluates exactly who he is now that he can no longer fight with a sword. He had pinned his entire identity onto his right hand, and now he has to find another identity.
All of this whilst everyone else in the world treats him like the exact same entitled scumbag that he used to be. He gets to see exactly what he had become through the eyes of someone else. This encourages all of the actions he takes later, as he tries to repent for the sins of his former life.
However, he is required to maintain a balancing act with his former persona, as it is shown that he can no longer fight at all, and manages to get by for a very long time using only the legend of his fighting prowess that had been built up around him for all of those years. He needs everyone to think that he is still the greatest swordsman alive so that no one will challenge him, as he will surely lose.
So in order to write the redemption story and allow Jaime's character arc to progress the way it did, the author was actually showing the readers that the character managed to get by on reputation alone, and could continue to do that whilst also attempting to correct all his former mistakes.