# Which subspaces of exterior power have decomposable bases?

Let $$V$$ be a real $$n$$-dimensional vector space, and let $$11$$. I wonder:

Is there a way to characterise which $$r$$-dimensional subspaces of the exterior power $$\bigwedge^k V$$ have bases which consist of decomposable elements?

Can we say "how rare" are such subspaces? Do they form a submanifold or an algebraic subvariety? (I think that they do form a closed subset of $$Gr_r(\bigwedge^k V)$$).

For $$r=1$$, this amounts to recognizing decomposable elements in $$\bigwedge^k V$$, which is what the Plucker relations do. But is there any sensible criterion for $$r>1$$?

Note that a subspace can admit a non-decomposable basis, even when there exist a decomposable basis: Take e.g. $$r=2$$ and

$$\text{span}(e_1 \wedge e_2+e_3 \wedge e_4,e_1 \wedge e_2-e_3 \wedge e_4)=\text{span}(e_1 \wedge e_2,e_3 \wedge e_4)$$.

One particular case where a subspace $$\tilde W \le \bigwedge^k V$$ has a decomposable basis, is when $$\tilde W= \bigwedge^k W$$ for some subspace $$W \le V$$. However, I don't know if there is an easy way to recognize subspaces of this form.