  In this paper, we consider all possible variants of Choi matrices of linear
maps, and show that they are determined by non-degenerate bilinear forms on the
domain space. We will do this in the setting of finite dimensional vector
spaces. In case of matrix algebras, we characterize all variants of Choi
matrices which retain the usual correspondences between $k$-superpositivity and
Schmidt number $\le k$ as well as $k$-positivity and $k$-block-positivity. We
also compare de Pillis' definition [Pacific J. Math. 23 (1967), 129--137] and
Choi's definition [Linear Alg. Appl. 10 (1975), 285--290], which arise from
different bilinear forms.
