I'm stuck on one part of a proof, and any help would be greatly appreciated!
Let $\omega:V \times V \to \Bbb{C}$ be a positive definite Hermitian form. Define $\omega_r: V \times V \to \Bbb{C}$ and $\omega_i:V \times V \to \Bbb{C}$ by $\omega(v,w) = \omega_r(v,w) + i\omega_i(v,w)$. Prove that $\omega_i$ is non-degenerate.
The parts I already have done are showing $\omega_r$ is positive definite and symmetric, and $\omega_i$ is skew-symmetric, but I can't figure out how to prove the non-degeneracy of $\omega_i$