The cross product in Euclidean space IR3 is an operation in which two vectors are associated to generate a third vector, also in space IR3. This product can be studied rewriting its basic equations in a matrix structure, more specifically in terms of determinants. Such a structure allows extending, for analogy, the ideas of the cross product for a type of the product of vectors in higher dimensions, through the systematic increase of the number of rows and columns in determinants that constitute the equations. So, in a n-dimensional space with Euclidean norm, we can associate n – 1 vectors and to obtain an n-th vector, with the same geometric characteristics of the product in three dimensions. This kind of operation is also a geometric interpretation of the product defined by Eckman . The same analogies are also useful in the verification of algebraic properties of such products, based on known properties of determinants.
P. Gritzmann and V. Klee, “On the Complexity of Some Basic Problems in Computational Convexity II. Volume and Mixed Volumes,” In: T. Bisztriczky, P. McMuffen, R. Schneider and A. W. Weiss, Eds., Polytopes: Abstract, Convex and Computational, Kluwer, Dordrecht, 1994, p. 29.