A multilinear form on a vector space V(F) over a field F is a map f:V(F)×...×V(F)->F such that c·f(u_1, ..., u_i, ..., u_n) = f(u_1, ..., c·u_i, ..., u_n) and f(u_1, ..., u_i, ..., u_n) + f(u_1, ..., u_i', ..., u_n) = f(u_1, ..., u_i + u_i', ..., u_n) for every c element F and any indexes i, j. For example, the determinant of a square matrix of degree n is an n-linear form for the columns or rows of a matrix.

bilinear form | vector space