Symmetric Multilinear mapping is zero

Question

Let $T:\underbrace{X\times X\times\dots\times X}_{n\ \text{times}}\to Y$ be a multilinear mapping that is symmetric. How can we prove that if:

$T(x,x,...,x)=0, \forall x\in X$

then $T\equiv 0$?

For $n=2$, I wrote $T(x,y)=1/4 \cdot(T(x+y,x+y)-T(x-y,x-y))$