I previously asked this on Mathematics Stack Exchange, to no result:

Consider the standard induced inner product structure on $\wedge^k\mathbb{R}^d$ given by defining $$\langle u_1\wedge \cdots \wedge u_k, v_1\wedge \cdots \wedge v_k\rangle:=\det ([\langle u_i, v_j\rangle]_{i,j=1}^d)$$ for simple $k$-vectors and extending to general $k$-vectors in the obvious fashion. If a linear subspace $W$ of $\wedge^k \mathbb{R}^d$ admits a basis of simple $k$-vectors, does its orthogonal complement have the same property?