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Why is cross product only defined in 3 and 7 dimensions? [duplicate]

Why $3$ and $7$? I know from some reading that Hurwitz's Theorem explains this, but can someone help me build some intuition behind this or perhaps provide a simpler explanation? It still seems ...
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Generalized Cross Product

I know that the cross product can be generalized as $$\text{cross}(x_0,...,x_{n-1})=\det\begin{vmatrix}&x_0&\\&x_1&\\&\vdots&\\e_1&\cdots&e_n\end{vmatrix}$$ where $e_i$ ...
The following vectors are linearly independent - $v1 = (1, 2, 0, 2)$ $v2 = (1,1,1,0)$ $v3 = (2,0,1,3)$ Find a fourth vector v4 so that the set { v1, v2, v3, v4 } is a basis fpr $\mathbb{R}^4$? I ...