  The definition of vector cross product (VCP) introduced by Eckmann only
exists in thethree- and the seven- dimensional vector space. In this paper,
according to the orthogonal completeness, magnitude of basis vector cross
product and all kinds of combinations of basis vector $\hat{e}_i$, the
generalized definition of VCP in the odd n-dimensional vector space is given by
introducing a cross term $X_{AB}$. In addition, the definition is validated by
reducing the generalization definition to the fundamental three- and
seven-dimensional vector space.
