vector product in general vector spaces

The vector product can be defined in any finite dimensional vector space $V$ with $dimV=n$. Let $v_1,\mathrm{\dots },v_n$ be a basis of $V$, we then define the vector product of the vectors $w_1,\mathrm{\dots },w_{n-1}$ in the following way:

$$w_1\times \mathrm{\dots }\times w_{n-1}=\sum _{j=1}^n{v}_{j}det(w_1,\mathrm{\dots },w_{n-1},v_j).$$

One can easily see that some of the properties of the vector product are the same as in ${ℝ}^3$:

• If one of the $w_i$ is equal to $0$, then the vector product is $0$.

• If $w_i$ are linearly dependent, then the vector product is $0$.

• In a Euclidean vector space $w_1\times \mathrm{\dots }\times w_{n-1}$ is perpendicular to all $w_i$.