It has been proved that cross product of vectors just is satisfied in 3D space and also in 7D and 8D spaces as specific cases. Besides, there are some other multiplication schemes of vectors such as wedge product, Ersatz cross product ... that are not quite the cross product (they do not satisfy the required conditions as in 3D space). On the other hand, in Dirac bra-ket notation of quantum mechanical vectors, the operators are introduced as outer (cross) product of a pair of vector. But it is not clear that this definition of cross product is the same and familiar cross product of vectors with the known required conditions or it is a specific case such as the others. And in principle, is there exist a definition of "cross product" of two vectors in infinite dimensional space that satisfies the conditions as in 3D space? I will gratefully appreciate if anyone could guide me.