>>14371518Cross product and Dot product predates the wedge product by 70 years. It's intended to produce a vector perpendicular to a pair of non-colinear vectors i.e. in . It just happens that it can be expressed through the determinant and can be expressed through the Hodge star operator composed with the wedge product.
Wedge product is not what you're looking as a generalization. At best, it's just an alternative way to calculate the dot product in conjunction with the Hodge star operator. Moreover, the wedge product can only perform a similar operation with vectors in -dimensional space since there are independent forms.
The actual generalization you're looking for is the Lie Bracket. A cross product is actual a type of Lie Bracket, skew-symmetric bilinear multiplication operation on a Vector Space. Not a type of wedge product. For example, for square matrices, is a Lie Bracket. For linear differential operators with smooth coefficients over the space of smooth functions real valued functions of n variables, the operation is another such Lie Bracket where is a function being acted upon by the operators, . The product of derivations on an algebra with the rule is another Lie Bracket (this is a generalization of the linear differential operators case).
