There exists a polar decomposition for matrices over the reals. What I would like to know is if an analog has been shown for groups of matrices over finite fields. If not, it would be great to get some feedback as to how to proceed, or some reasons why it may not exist.

Specifically, I'm interested in decomposing invertible matrices over $\mathbb{F}_q$ as a product $x = s.o$ where $s$ is symmetric and $o$ is orthogonal.