Wedge product with a non-degenerate form

Question

Let $\alpha$ be a non-degenerate form in $\Lambda^k(V)$ for some vector space $V$, $\dim V = n$. (Here non-degenerate means that if $x\in V$ is nonzero, then $(y_1 , ... , y_{k-1}) \mapsto \alpha(x , y_1 , ... , y_{k-1})$ is a nonzero form). Is it true that if $\beta$ is a nonzero $\ell$-form for $\ell + k\leq n$, then $\alpha \wedge\beta \not=0$? Is this true under stricter conditions on $\beta$ (e.g. $\beta$ also non-degenerate)?

Answer 1

Let us take $n=4$ and $$ \alpha:=e^1\wedge e^2+e^3\wedge e^4\text{ and }\beta:=e^1\wedge e^3+e^2\wedge e^4. $$ Both $\alpha$, $\beta$ are non-degenerate forms but $$ \alpha\wedge\beta=0. $$