Cross product in $\mathbb R^n$ (from Spivak's book) Spivak defines cross product in this way:

$\quad$ We conclude this section with a construction which we will restrict to $\mathbf{R}^n$. If $v_1,\ldots,v_{n-1}\in\mathbf{R}^n$ and $\varphi$ is defined by $$\varphi(w)=\det\pmatrix{v_1 \\ \vdots \\ v_{n-1} \\ w},$$ then $\varphi\in\Lambda^1(\mathbf{R}^n)$; therefore there is a unique $z\in\mathbf{R}^n$ such that $$\langle w,z\rangle=\varphi(w)=\det\pmatrix{v_1 \\ \vdots \\ v_{n-1} \\ w}$$ This $z$ is denoted $v_1\times\cdots\times v_{n-1}$ and called the cross product of $v_1,\ldots,v_{n-1}$.

Why such a $z$ exists and why is it unique? When solving problems involving this notion, how do I find this $z$ explicitly (if it's possible)?
Also, what's the meaning of this cross product? Many sources say that the usual cross product in $\mathbb R^3$ can't be generalized to higher dimensions.

It exists because the inner product $\langle \cdot,\cdot\rangle$ is non-degenerate and we apply the Riesz representation theorem (whose proof in the finite dimensional case is easy) for the linear functional $\varphi\colon \Bbb R^n \to \Bbb R$ given by $$\varphi(w) = \det(v_1,\ldots,v_{n-1},w).$$The Riesz vector $z$ satisfies $\langle w,z\rangle =\varphi(w)$ for all $w \in \Bbb R^n$, and is then denoted by $v_1\times\cdots\times v_{n-1}$. If you write $$v_1\times\cdots \times v_{n-1} = \sum_{i=1}^n \langle e_i,z\rangle e_i$$by orthonormal expansion and compute $\langle e_i, z\rangle = \det(v_1,\cdots,v_{n-1},e_i)$ using Laplace expansion, you'll get a formal expression for $v_1\times\cdots \times v_{n-1}$ with a determinant just like the $3$-dimensional case. For example, in $\Bbb R^4$ we have $$e_1\times e_2 \times e_3 = \begin{vmatrix} e_1 & e_2 & e_3 & e_4 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{vmatrix} = e_4,$$and so on.
