Find two bilinear forms with the same quadratic form over $\mathbb F_2$

Question

Let $V$ be a $K$-vectorspace with a bilinear form $\langle , \rangle$ and the associated quadratic form $q:V \to K, v \mapsto \langle v,v \rangle$.

Let $K = \mathbb F_2$. Are there two different bilinear forms with the same quadratic form over this field? I cannot come up with an idea.

Answer 1

Edit: Consider $V=K^2,\ b_1(\mathbf u, \mathbf v)=\mathbf u^T\pmatrix{0&1\\ 0&0}\mathbf v$ and $b_2(\mathbf u, \mathbf v)=\mathbf u^T\pmatrix{0&0\\ 1&0}\mathbf v$.

Write $\mathbf u^T=(x,y)$ and $\mathbf v^T=(z,w)$. Then $b_1(\mathbf u, \mathbf v)=xw\ne yz=b_2(\mathbf u, \mathbf v)$ in general but $b_1(\mathbf u, \mathbf u)=b_2(\mathbf u, \mathbf u)=xy$.