Canonical forms of the natural action of $O(n,\mathbb{C})$ on $sym(n,\mathbb{C})$

Question

I want to learn about the natural action of the complex orthogonal group, $O(n,\mathbb{C}):=\{g\in M(n,\mathbb{C}) : gg^T=\mbox{Identity}\}$, on the complex vector space of complex symmetric $n\times n$ matrices, $sym(n,\mathbb{C}):=\{X \in M(n,\mathbb{C}) : X = X^{T}\}$; $$ X \mapsto gXg^{-1}=gXg^{T};\, X\in sym(n,\mathbb{C}) \mbox{ and } g\in O(n,\mathbb{C}). $$ Is there any canonical form for the above mentioned action? How could I understand the case n=3? Thank you.