There are numerous applications of symmetric bilinear forms: They show up in the Hessian matrix in optimisation, for instance. What I'm curious about is where asymmetric bilinear forms occur.
I know that the Fundamental Matrix in Computer Vision is one such application.
An asymmetric bilinear form is a map $$B: U \times V \to k$$ linear in both its arguments, and where $k$ is a field. We will exclude examples where $U = V$ and $B$ is symmetric (i.e. $B(u,v)=B(v,u)$ for all $u, v \in V$).
The determinant of two-by-two matrices is a bilinear form, when seen as a function of the two columns of the matrix. It is anti-symmetric. $$ \Bigl(\pmatrix{v_1, v_2}, \pmatrix{w_1, w_2} \Bigr)\mapsto v_1 w_2 - v_2 w_1. $$ Any bilinear form is the sum of a symmetric bilinear form and an anti-symmetric bilinear form.
A symplectic form is a non-degenerate antisymmetric bilinear form that exists on all even dimensional spaces. It occurs in hamiltonian mechanics and of course in symplectic geometry.