How to genarelize the vector product in three dimensions to higher dimensions?

[wdlang]

it seems that the vector product between vectors in three dimensions is peculiar property of the three dimensional space

 

[Landau]

See here.

 

[wofsy]

it seems that the vector product between vectors in three dimensions is peculiar property of the three dimensional space

That is true.

If one has a metric you can talk about volumes and hypervolumes and come up with an a hypercube orthogonal to any given hypercube of complementary dimension. If you have an orientation then an oriented hypercube will have an oriented complement.

This all can be done with wedge product one you have an orthonormal basis.