suppose i'm asked to find the associated bilinear form for the quadratic form $q(v)=x_1x_3+x_2^2,$ wonder if i first need to find the associated matrix to this quadratic form, then in order to get the associated bilinear form i'd do $f\big((x_1,x_2,x_3),(y_1,y_2,y_3)\big)=(x_1~x_2~x_ 3)[\text{matrix of the quadratic form}](y_1~y_2~y_3)^t$ ?
You want $\begin{pmatrix}x_1 & x_2 & x_3\end{pmatrix}\begin{pmatrix}a & b & c \\ b & d & e \\ c & e & f\end{pmatrix}\begin{pmatrix}x_1 \\ x_2 \\ x_3\end{pmatrix}= x_1x_3+ x_2^2$ for all $x_1$, $x_2$, and $x_3$.