A quick way to generate 3x3 matrices with determinant equal to 1? Perhaps a formula involving the row number and column number of an element or just some parametric equations for each element. I know that I can just multiply two of these matrices together to get another but I'd rather not do that. Also, I wouldn't want too many zeroes in the matrix (1 or 2 at most).

If you choose all matrix elements except one to be uniformly random (say, floating point numbers between 0 and 1, which many programming languages will do for you), then it is esssentially probability 0 that you will get a subfactor determinant that equals 0. So just expand the determinant along the first row, and choose all elements random except the last element in the first row, and then choose the last element in the first row deterministically so that when you compute the determinant expanded along the first row you get 1. This is not a very nice random distribution on matrices in a certain sense but it is definitely a particular random distribution on the set of all $3 \times 3$ matrices with determinant 1.
