For $k$ a field with $\text{char}(k)\neq 2$ we have a bijection from the quadratic forms on a vector spave $V$ to the symmetric bilinear forms $V\times V\to k$, namely, $q\mapsto b_q$ where $b_q(v,w)=q(v+w)-q(v)-q(w)$.
What goes wrong, if $\text{char}(k)=2$?
If your mapping to bilinear forms is $b_{q}(v,w)=q(v+w)−q(v)−q(w)$, then your inverse mapping is $q_{b}(v)=\frac{1}{2}b(v,v)$, where $\frac{1}{2}$ is defined to be the multiplicative inverse of $1+1$, where $1$ is the multiplicative unity in $k$. But if $k$ has characteristic $2$, then $1+1=0$ has no multiplicative inverse.