Linear algebra lemma

First note that if a 2-form is degenerate, it is 0 on some 4-subspace (take a lagrangian subspace of the quotient by the kernel).

Now, assume not. Pick two elements that span $W$. If either of them has 4-d kernel, it is 0 on any 4-d subspace, and we can use whichever on the other vanishes on.

Thus, every element of $W$ has 2-d kernel. If two elements had different kernels, then one of their linear combinations would have no kernel. Thus, they all kill the same 2-d subspace. Thus, we've reduced to the statement that any two 2-forms on a 4-d space $Z$ have a common Lagrangian subspace. Pick any line $L$; this is isotropic for both, since all lines are. Consider the intersections of the symplectic orthogonals of $L$ for the two 2-forms. These are 3-d, so their intersection is a 2-space. Now you win.