For a symmetric or antisymmetric bilinear form $\varphi$ on a vector space $V$, if $\varphi(x,y)=0$ for all $x,y\in V$ then also $\varphi(y,x)=0$.

I was wondering if this is also a necessary condition for this to happen.

For a symmetric or antisymmetric bilinear form $\varphi$ on a vector space $V$, if $\varphi(x,y)=0$ then also $\varphi(y,x)=0$ ($x,y\in V$).

I was wondering if this is also a necessary condition for this to happen, that is whether a form with this property must be either symmetric or antisymmetric.