Let $ V$ be a vector space over the finite field $ \mathbb{F}_q$ of dimension $ n\geq 6$ . Let $ \omega\in\bigwedge^{n-3}V$ be a nonzero element. Define the anihalator subspace of $ \omega$ by $ V_\omega=\{z\in V: \omega\wedge z\neq 0\}$ . It is well known that $ 0\leq \dim V_\omega \leq n-3$ . Suppose for this given $ \omega$ we have $ \dim V_\omega= n-k$ where $ k\geq 5$ . I want to prove that $ $ |\{x\in V: \omega\wedge x\in \bigwedge^{n-2}V\text{ is decomposable}\}|\leq q^{n-k}(q^{k-3}-1)(q^2+1). $ $

By a decomposable element of $ \bigwedge^{d}V$ , we mean a nonzero element that can be written as $ v_1\wedge\ldots\wedge v_k$ for some linearly independent elements $ \{v_1,\ldots,v_k\}$ of $ V$ . I tried that and could prove this when $ k=6$ but I am not able to prove this in general.