Given a bilinear form, $B : U \times V \rightarrow K$ we show how we can represent it with a matrix, with respect to a particular pair of bases for $U$ and $V$ Suppose $U$ and $V$ are finite-dimensional and we have chosen bases, ${{\cal B}_1} = \{e_1, \ldots\}$ and ${{\cal B}_2}=\{f_1, \ldots\}$ Now we define the matrix $C$ with entries $C_{ij} = B(e_i, f_j)$ This will be the matrix associated to $B$ with respect to this basis as follows; If we write $x,y \in V$ as column vectors in terms of the chosen bases, then check $B(x,y) = x^T C y$ Further if we choose the corresponding dual bases for $U^\ast$ and $V^\ast$ then $C$ and $C^T$ are the corresponding matrices for $B_R$ and $B_L$ respectively (in the sense of linear maps). Thus we see that a symmetric bilinear form is represented by a symmetric matrix, and similarly for skew-symmetric forms.

Let ${{\cal B}_1^\prime}$ and ${{\cal B}_2^\prime}$ be new bases, and $P$ and $Q$ the corresponding change of basis matrices. Then the new matrix is $C^\prime = P^{T}CQ$