The wedge product and cross product are both antisymmetric operations on vectors, but they yield different mathematical objects: the wedge product results in a bivector, while the cross product produces a pseudo vector. The wedge product is defined for any vector space, existing in the space of 2-forms, denoted /\^2(V). In three-dimensional space, the bivector space is isomorphic to the vector space, leading to the identification of the wedge product with the cross product. This identification is non-canonical, meaning it depends on the specific context of three dimensions. Understanding these distinctions is crucial for grasping their applications in geometry and physics.