Let $\phi$ be a bilinear form over a linear space $V$ (linear space over a field $F$). A vector $v\in V$ is said to be isotropic (with relation to $\phi$) if $\phi(v,v)=0$. Let $u,v\in V$. Show that $u\pm v$ are isotropic $\iff \phi(u,v) = -\phi(v,u)$.
The ($\Rightarrow$) part is ok, I'm stuck with showing the reverse. My attempt: I already showed that $\phi(u+v, u+v) = \phi(u-v, u-v) = \phi(u,u) + \phi(v,v)$ but this leads to nothing as far as I can see. I'll enjoy any help.