In [Space-Time-Matter](https://www.gutenberg.org/ebooks/43006) Hermann Weyl claims that had we lived in a world of more than three spacial dimensions, we would have known all along that quantities such as angular momentum are skew-symmetric tensors, and the "error" of using the three-component "vector product" would never have occurred. Pages 66-67.

His first example of a skew-symmetrical(sic) tensor is angular momentum:

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> In mechanics the skew-symmetrical tensor product of two vectors occurs— 1. As moment of momentum (angular momentum) about a point $\mathcal{O}.$ If there is a point-mass at $\mathcal{P}$ and if $\xi^1,\xi^2,\xi^3$ are the components of $\vec{\mathcal{OP}}$ and $u^{i}$ are the (contra-variant) components of the velocity of the points at the moment under consideration, and m its mass, the momentum of momentum is defined by
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> $$L^{ik}=m\left(u^{i}\xi^{k}-u^{k}\xi^{i}\right).$$
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Why/how should we define something called *angular momentum* as a skew-symmetric tensor without first learning about it as a cross-product?

I'm no expert in the application of her theorem, but it may be worth pointing out that (IIRC) Emmy Noether was Weyl's graduate student.