Are there pseudovectors in more than three dimensions?

Question

Do pseudovectors only exist with vector product in $\mathbb R^3$ or do they appear in higher dimensions too?

Answer 1

In $\mathbb R^n,$ given $n-1$ linearly independent vectors, you can construct a new vector that can be considered to be the cross product of those vectors. That is then a pseudovector.

If the $n-1$ vectors are $u^{(1)} = (u^{(1)}_1, \ldots, u^{(1)}_n), \ldots, u^{(n-1)} = (u^{(n-1)}_1, \ldots, u^{(n-1)}_n)$ then the pseudovector is given formally by $$ \begin{vmatrix} u^{(1)}_1 & \cdots & u^{(1)}_n \\ \vdots & & \vdots \\ u^{(n-1)}_1 & \cdots & u^{(n-1)}_n \\ \hat e_1 & \cdots & \hat e_n \end{vmatrix} $$ where $\hat e_k$ is the unit vector in the direction of the $k$:th coordinate.