Let $v\in \mathbb{C}^n$ be an $n$ dimensional complex vector. Define the non-standard bilinear form $\left< u,v \right> = u^T v$ (the usual inner product except without the conjugation). What are the properties of the self orthogonal vectors, those such that $\left < v,v \right>=0$? What are the properties of this space of vectors?

For example, if we say that $v^Tv = 0$ has the null property, then so does $\lambda v$ for any complex $\lambda$, and so does $v^*$ (the conjugated vector). Clearly, the set of null vectors is not a vector space since generally $u+v$ is not null if $u$ and $v$ are null.

This may be well known, but I just have no idea what to Google for. Any pointers to existing literature are appreciated.