By duality, this is equivalent to a function formula_124 (fixing any two inputs gives a function formula_125 by evaluating on the third input) and in the presence of an inner product (such as the dot product; more generally, a non-degenerate bilinear form), we have an isomorphism formula_126 and thus this yields a map formula_127 which is the cross product: a (0,3)-tensor (3 vector inputs, scalar output) has been transformed into a (1,2)-tensor (2 vector inputs, 1 vector output) by "raising an index".