Surjectivity of an alternating, non degenerate bilineair form

Question

I have a function $\varphi: V \times V \to k$ where $V$ is a vector space over ground field $k$ (although I would also like to prove this for $R$-modules). I know that $\varphi$ is alternating, that is $\varphi(v,v) = 0$ for all $v$, and non degenerate, that is if $\varphi(v,w) = 0$ for all $w$ then $v$ is the zero vector. ($\varphi$ is symplectic)

Is it possible to conclude from these facts that $\varphi(v,-): V \to K $ is surjective for all $v$? Are there other assumptions that wil make $\varphi(v,-): V \to K $ surjective?

Answer 1

Yes, this map is always surjective for $v\neq 0$. Find a $w$ such that $\varphi(v,w)\neq 0$, then you can obtain any element $x$ of $K$ by plugging in $(x/\varphi(v,w))w$.