# Does every subspace of the exterior algebra of dimension $>1$ contain a decomposable element?

Let $$V$$ be a real $$n$$-dimensional vector space, and let $$W \le \bigwedge^k V$$ be a subspace . Suppose that $$\dim W \ge 2$$. Does $$W$$ contain a non-zero decomposable element?

If $$\dim W=1$$, then clearly we can take $$W=\text{span} (\sigma)$$ for some non-decomposable $$\sigma \in \bigwedge^k V$$.