i hate to be "that guy"... you know, the one that asks all those questions that could be solved with just a lil' bit of experimentation, but i have no patience, so here goes:
i always hear people complaining about knee pain whilst brakeless fixie riding, but i've never had that problem.
what i DO have is pain in my left ankle from skids. if you were looking at a bike buisness-side-out, i skid by lifting my left foot at 3 o'clock, and pushing back with the right at 9 o'clock, which is making my left ankle sore as hell from pulling up on the clips.
Ive never heard of ankle pain from skidding...but I guess anything is possible. Maybe try lifting with your right foot? If not maybe try stretching or loosening up your ankles before you ride...if such a task is even possible.