nondegenerate alternating bilinear forms

Question

Let $V$ be a finite dimensional vector space over $F$. Let $B_1$ and $B_2$ be two nondegenerate alternating bilinear forms. For $i=1,2$ define $$S_i=\lbrace \sigma\in GL(V)\text{ }|\text{ }B_i(\sigma(x),\sigma(y))=B_i(x,y) \text{ for all }x,y\in V\rbrace$$

How to prove that $S_1$ and $S_2$ are conjugates in $GL(V)$ ?

Answer 1

Hint: Since $B_i$ is non degenerated, you can find a basis $(e^i_1,..,e^i_n,f^i_1,..,f^i_n)$ such that $B(e^i_j,e^i_k)=B(f^i_j,f^i_k)=0, B(e^i_j,f^i_k)=\delta_{jk}$.

Let $g(e^1_j)=e^2_j, g(f^1_j)=f^2_j, S_1=gS_2g^{-1}$.