Show that $\varphi(x,ix)=0 $ iff $\varphi$ is the real part of a complex inner product in $E$.

Question

Let $\varphi$ be a real inner product $\varphi:E \times E \to \mathbb{R}$ in a complex vector space $E$. Show that $\varphi(x,ix)=0 \ (\forall x\in E) $ iff $\varphi$ is the real part of a complex inner product in $E$.

I found it easy enough to show the if part, but I have no clue on the only if. Any hints?