I'm doing self taught mathematics on Khan Academy, and just finished geometry and am starting Algebra II. I don't really do the quizzes and tests, but feel like I know the stuff really, really, really well. Am I doing something wrong?

You should do a few of them just for the feedback, as others have mentioned. I’ve heard others say this and I agree: the best way to learn math is by doing many questions (at an appropriate difficulty level with some feedback so that you can clear up misconceptions). Once you feel like they’re getting too easy, try harder ones.