Find a minimal subspace $W_{\alpha}$ in $V$ for a given $\alpha \in \bigwedge^{p}(V)$ for which $\alpha \in \bigwedge^{p}(W_{\alpha}).$

Question

Let $V$ be a vector space of dimension $n$, and $1\leq p \leq n$, show that for every $\alpha \in \bigwedge^{p}(V)$ there exists a minimal subspace $W_{\alpha}$ in $V$ such that $\alpha \in \bigwedge^{p}(W_{\alpha})$. I was thinking about using induction, maybe contracting $\alpha$ along some vectors could be of help.