Properties of k-linear form

Question

I would like to know if this is correct:

Let be $f:V\times V\times\text{ }...\times V\rightarrow W $ a symetric k-linear form. If $f(v,..,v)=f\cdot v^k=0,\forall v\in V$, then $f=0$.

I proved this for k=2, but I don't know if it's worth it in general.