Cross Product in Higher Dimensions

I had a teacher in multivariable calculus who said the cross product only works for three dimensions. That bothered me, so I started working on extending the definition of the cross product to higher dimensions. In general, my definition uses the determinant method via cofactor expansion, where an n x n matrix is constructed with n-1 linearly independent vectors in R^(n). I'm trying to expand it to all dimensions n>1 using induction, and I've successfully proven it works for n=2 and n=4, but the more I look into finishing the proof the more difficult it feels.

From there, see the following screenshot because I can do the type setting better there than here.

https://preview.redd.it/upa1xgj39z3h1.png?width=1017&format=png&auto=webp&s=ab2fb8b92de8f570bfa9cac4cb1b5b79ff3c7ba5

And there I'm stuck. I can't figure out how to even approach finding those scalars c. Everywhere I look, it seems the only way to come at it is through brute force, which isn't helpful in induction. Here are some of my questions:

  • Does anyone know of some theorem I can use to put this mess into a better format? Something about cofactors, perhaps?
  • Is there another definition of the cross product that I'm missing? Perhaps a different definition would be easier to use.
  • Is it possible to compute the determinant for an arbitrary, and arbitrarily large, matrix? In my research along those lines, I mostly found things about LU factorizations and such, but I can't meaningfully use the LU factorization of an arbitrary matrix. And I know using the cofactor expansion method scales in computation length with n!, so I feel like I need another way to go about this.
  • Is this impossible to prove? It feels like it's tied up with the P vs. NP problem, and that's obviously beyond my ability to solve.

The following screenshots are my computations for n=2 and n=4, in case it's helpful. Thank you in advance for your thoughts and suggestions!

https://preview.redd.it/wgcot89k9z3h1.png?width=725&format=png&auto=webp&s=4d7649fd8c8c5a56843dcd4c735f68ef2cab0656

https://preview.redd.it/avxx9j8k9z3h1.png?width=647&format=png&auto=webp&s=1eec75553f9b4ca1f11704e2d88f1aa52718eb02

EDIT: I thought it was clear from my explanation already, but yes, I do understand that my proposed formulation of the cross product is not a binary operation. Frankly, I don't really care. I'm just exploring and trying to extend a certain procedure for generating orthogonal vectors.

Author: Gringo256