Suppose that ω is a linear 2-form on a vector space V . Prove that there exists a basis $e_1, . . . , e_r, f_1, . . . , f_r, h_1, . . . , h_n$ such that ω is expressible as
$ω = e_1^*∧f_1^*+e_2^*∧f_2^*+...e_k^*∧f_k^*$
My attempt : I divided V into the subspace W on which ω is degenerate and K on which it is nondegenerate. Let a basis for W be $h_1, . . . , h_n$ which we leave aside.
For K, I know that ω is now a symplectic form(as nondegenrate). I am told that it has something called a symplectic basis. Can anyone elaborate on this and finish the proof?