Let $(M,g)$ be a compact Riemannian manifold. Assume that $\omega \in \Omega^p(M)$ and $\omega'\in \Omega^q(M)$ are $p$ and $q$-forms with constant and non-null norms, i.e, $$\|\omega(p)\| = C > 0,~\|\omega'(p)\| = D > 0,~\forall p \in M.$$
One can assume that $\omega$ and $\omega'$ are closed. Is there a sufficient condition in order to $$\max_{p\in M}\|\omega\wedge\omega'\| \neq 0?$$
I was expecting that some formula like $$\|\omega\wedge \omega'\|^2 = \|\omega\|^2\|\omega'\|^2 - \langle \omega,\omega'\rangle^2$$ would work (is this formula true?).
In this case, since $\omega$ and $\omega'$ are closed, $\langle \omega,\omega'\rangle$ would be constant. Therefore, if there is a point $p\in M$ such that $\cos^2\varphi\neq 1$ one will have the claim, where $\varphi$ is the angle between $\omega$ and $\omega'$.
What do you think?