I've been reading about skew-symmetric bilinear maps, and the following question came to my mind randomly. Unfortunately, I couldn't come up with a satisfactory explanation to convince myself.
Assumptions: Let $V$ be an $n$-dimensional real vector space (with $n>0$) and consider a bilinear map $\phi:V\times V\to \mathbb{R}$. Assume that $\phi$ is not the zero map and that it is skew-symmetric (i.e., $\phi(u,v)=-\phi(v,u),$ for all $v,u\in V$). Let $\alpha\in V$ such that $\phi(\alpha,v)=0,$ for all $v\in V.$
My question is, is it necessarily true that $\alpha=0$? If not, are there any counterexamples to this claim?
Thanks in advance.