I learned today that the set of symmetric bilinear forms (SBF) of the form $\sigma: V\times V \rightarrow F$ is in one-to-one correspondence with the set $\mathcal{S}_n(F)$ of $n$ by $n$ symmetric matrices, where $V$ is a finite-dimensional vector space over an infinite field $F$. But by a theorem I learned previously, $M_n(F)$ (of which $\mathcal{S}_n(F)$ is a subset) is in one-to-one correspondence with $\mathcal{L}(V)$, the set of all linear operators on $V$. In both cases, the correspondence is made between the linear operator/SBF and its matrix representation.
Then, presumably, with respect to a particular basis $B$ of $V$, a symmmetric matrix $A \in \mathcal{S}_n(F)$ will represent both a linear operator $L$ and a SBF $\sigma$. Besides having the same matrix representation (which maybe is as 'deep' as it gets), is there anything else I can say regarding the connection between the $L$ and $\sigma$ which satisfy $[L]_B = [\sigma]_B$?
Thanks in advance!