Is the cross product in spaces of higher dimensions useful in this way?

Question

Looking at the Stack Exchange question "Four dimensional cross product of THREE vectors" I sense that in $\mathbb{R}^n$ I can define a cross product between $n-1$ vectors that gives a vector orthogonal to them. A first question: how I transform $|\mathbf{A} \times \mathbf{B}|=|\mathbf{A}| |\mathbf{B}| \sin \theta$ in this context? But what really matters: can I exploit this extension of cross product to find the $n$-volume "generated" by $n$ linear independent vectors calculating $\mathbf{V}_1\cdot(\mathbf{V}_2 \times \mathbf{V}_3 \times \dots \times \mathbf{V}_n)$? And given a $\left\{ \mathbf{E}_i \right\}$ basis in $\mathbb{R}^4$, can I find its dual by calculating $\mathbf{E}^1 = \frac{ \mathbf{E}_2 \times \mathbf{E}_3 \times \mathbf{E}_4 }{\mathbf{E}_1\cdot( \mathbf{E}_2 \times \mathbf{E}_3 \times \mathbf{E}_4)}$ and so on? This could be exploited to show that in rectilinear (but in general not uniform and not orthogonal) coordinates, both covariant and contravariant basis vectors are everywhere constants.