equivalent characterization of anti-symmetric operator in complex Hilbert space

Question

Assume that $\mathcal{H}$ is a complex Hilbert space and $B$ is a bounded linear operator such that $B^* = -B$. In some notes I found the claim that $$B^* = -B \iff \forall h \in \mathcal{H},~ \langle Bh,h\rangle = 0.$$ The author says this equivalence follows by the polarization identity. After spending hours on it I still could not figure out how this equivalence is derived (remark: it is fairly easy to see that the real part of $\langle Bh,h\rangle$ equals $0$ for all $h$ if $B$ is anti-aymmetric, but I do not know how to treat the imaginary part of $\langle Bh,h\rangle$) Thanks!