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.
$$\omega(X,X)=0 \ \ \forall \ \ \ \ X \implies \omega(X,Y)=0\ \ \ \forall \ \ \ X,Y$$
Thanks in advance.

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