Cross Product of N vectors

Today, we are going to define the cross product of ${\displaystyle N}$ ${\displaystyle \mathbb {A} ^{N+1}}$vectors in this blog post, where ${\displaystyle \mathbb{A}}$ is any field where multiplication is commutative and outputs a member in ${\displaystyle \mathbb{A}}$ itself

The definition we use will be the determinant of an ${\displaystyle N+1}$ x ${\displaystyle N+1}$ matrix with the first row containing the labels of the coordinates and the next ${\displaystyle N}$ rows containing the vectors themselves.

Conclusions:

From this, we can conclude that the cross product of a vector with components ${\displaystyle (a_{1},a_{2})}$ is ${\displaystyle (a_{2},-a_{1})}$. If the cross product is the zero vector, we can conclude that the vector involved is the zero vector itself.