Simple question on symmetric tensors

This question seems to be silly, but i am really confused. Suppose we have a symmetric $2$-tensor $\omega$, I want to prove
$$\omega(X,Y)=0\ \ \ \forall \ \ \ X,Y \iff \omega(X,X)=0 \ \ \forall \ \ \ \ X$$
The first direction is easy. The problem is the second direction i.e.
$$\omega(X,X)=0 \ \ \forall \ \ \ \ X \implies \omega(X,Y)=0\ \ \ \forall \ \ \ X,Y$$
This is the polarization identity. Given that $\omega(X,X)=0$, we get that $\omega(X+Y,X+Y)=0=\omega(X,X)+\omega(X,Y)+\omega(Y,X)+\omega(Y,Y)=2\omega(X,Y)$. Then assuming the characteristic of the ground field is not $2$, we get that $\omega(X,Y)=0$, for all $X, Y$.