5 dimensional angles (not 2D angles in 5 dimensions)

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Given $2$, $2D$ vectors we can calculate the angle inside these vectors using either dot or cross product. - (And presumably many other methods too)

Given $3$, $3D$ vectors, how would I calculate the solid angle inside them?

Given $4$, $4D$ vectors, how do I calculate the 4-angle inside of them?

And then finally, $5$, $5D$ vectors same question.

My problem is actually in 5 dimensions, so that's the one I'm most interested in. Though it would be lovely to see a pattern emerge from calculation to calculation.

Thank you in advance