I am trying to prove the following from a book I am reading through.
Thm: If $\omega$ is a 2-form on $\mathbb{R}^4$ and $\omega \wedge \omega = 0$, then $\omega$ is decomposable. Note decomposable means can be written of the form $\omega = x \wedge y$.
I know that I can write my 2-form as $\omega = a_1 e_1 \wedge e_2 + a_2 e_1 \wedge e_3 + a_3 e_1 \wedge e_4 + a_4 e_2 \wedge e_3 + a_5 e_2 \wedge e_4 + a_6 e_3 \wedge e_4$, where $e_1, e_2, e_3, e_4$ are basis elements. Then I have $0 = \omega \wedge \omega = (a_1a_6-a_2a_5+a_3a_4)(e_1 \wedge e_2 \wedge e_3 \wedge e_4)$. Thus $(a_1a_6-a_2a_5+a_3a_4)=0$.
I don't know what to do from here. Thanks for the help!