Does there exists a vector space $V$ and a bilinear form $w$ on $V\oplus V$ such that $w$ is not identically zero but $w (x,x) =0$ for every $x \in V$?
My work is : if $M_2$ spanned by $\{(1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1)\}$, and the bilinear form $w$ is defined by $w (x,x) = x_1y_2 +x_2y_1$ , then $w(x,x) = 0$ for every $x\in M_2$.
Am I correct?
My question is : How to find another vector space which is not a matrix space satisfies the above condition?
First of all, in your work, I suppose that you have defined $w$ such that $w(x,y) = x_1y_2+x_2y_1$, correct? In that case, pick
$$x = \begin{pmatrix}1 & 1 \\ 0 & 0\end{pmatrix} \in M_2 \, ,$$
and you have $w(x,x) = 1+1 = 2 \neq 0$.
Since you only required that $w$ was supposed to be a bilinear form (and not necessarily an inner product), we can choose any vector space $V$ together with the bilinear form $w(x,y) = 0$ for all $x,y\in V$.
If you want a nontrivial example, let's look at the vector space $\varDelta = \left\{\begin{pmatrix}x \\ x\end{pmatrix} \mid x\in \mathbb R\right\} \subset \mathbb R^2$, being the line $y = x$ through the origin in $\mathbb R^2$ together with the bilinear form $w(x,y) = x_1-y_2$ for example. Then we obviously have
$$w(x,x) = x-x = 0 \quad \forall x\in \varDelta$$