Classification of Matrices and normal forms

Question

As is shown in couses of Linear Algebra, for every square matrix $A$ one can choose $S,T,P\in GL(n,K)$ so that $SAT^{-1}=\operatorname{diag}(1,...1,0,...,0)$ and $PAP^{-1}$ is in Jordan normal form. But what if $S,T,P$ are also restricted to being orthogonal? Are there also special forms $A$ can be transformed to then? And what about bi-linear forms? It is shown that with $S\in GL(n,K)$, one can transform the matrix representation $A$ of a symmetric bi-linear form to $SAS^T=\operatorname{diag}(1,...,1,-1,...,-1,0,...,0)$, but what about non-symmetric bi-linear forms, orthogonal $S$ or transformations like $SAT^T?$ I know that's a lot of questions, I'd also be very happy about partial answers or helpful links.