If the characteristic of the ground field is not 2, then a bilinear form B on a vector space V can be written uniquely in the form B = A + S, where A is an antisymmetric bilinear form and S is a symmetric bilinear form. Then you will generally have invariants of the pair to deal with. For example, there will be the ranks of A and S and, for S, the type of the quadratic form. In a generic case, for example, when S is nondegenerate, one can then turn A into a linear map L from V into itself via the rule A(x,y) = S(x,Ly) and then the characteristic polynomial of L as a linear map will provide invariants of the original B. Any canonical form will have to be rich enough to capture such invariants.