A Particular Symmetric Bilinear Map Does there exist symmetric bilinear map on $\mathbb{C}^n$ such that 

$$<v, v> \neq 0 \quad \forall v \in \mathbb{C}^n-\{0\}?$$


Nope. E.g. bilinearity means linearity in the first argument and thus:
$$
\langle 0, 0 \rangle
= \langle 0\cdot 0, 0 \rangle
= 0 \langle 0, 0 \rangle
= 0
$$
and $0 \in \mathbb{C^n}$.
