Intuitive

The exterior product of two vectors represents the (oriented) area of the parallelogram enclosed by the two vectors. The exterior product of three vectors represents the (oriented) volume of the parallelepiped enclosed by the three vectors.

The exterior product generalizes the cross product and (scalar) triple product in such a way that we can calculate area and volume elements in higher dimensional spaces.

The exterior product is also known as Grassmann product or wedge product.

Concrete

The exterior product is calculated by taking the tensor product and antisymmetrizing the result. The picture below shows how to do that for two 3D vectors (top), three 3D vectors (center), and a 3D vector and an antisymmetric tensor (bottom). The relationship to the cross product, the (scalar) triple product, and the dot product is shown in red. For a more detailed explanation of this picture see Fun with Symmetry.

Abstract

Why is it interesting?