Symmetric Bilinear form B clearly determined by the images of B(v,v)

Question

Let $V$ be a $\mathbb{K}$-Vectorspace and $B : V×V \rightarrow \mathbb{K}$ a symmetric Bilinearform. Show that $B$ is clearly determined by $B(v,v)$ for $v\in V$.

I took an Element $(v,w)$ of $V$ and tried to split its image $B(v,w)$ into a sum where both entrys in the tupels are the same $"(x,x)"$-form. That however didn't work.

I would appreciate some hints on how to tackle this problem.

Answer 1

I assume that the characteristic of $|mathbbK$ is not $2$, $B(v+w,v+w)-B(v,v)-B(w,w)=B(v,v)+B(v,w)+B(w,v)+B(w,w)-B(v,v)-B(w,w)$

$=2B(v,w)$.