Er, could you clarify the question? Although I don't know precisely what you're asking, I suspect the answer has to do with the Quaternions and Octonions.
If you mean 'why do their exist anti-symmetric linear pairings x/\y : R^nxR^n-->R^n for some n, and not others', then Hurkyl is getting there. There is a theorem in differential geometry that explains this, though I don't know what it is saying (i.e. I can't encapsulate it into a nice bite sized slogan for the lay person).
existence of pairings produces vector fields on the sphere, and these exist only in a few cases. maybe this is related.
Well, the idea is that a matrix is created. The determinent can do very funny things. Just try to find the determinents of 3x3, 4x4, 5x5, 6x6, 7x7. You may figure out why...
PLease could you elaborate on why determinants of matrices can explain the (non-) existence of smooth (no-where zero, I imagine) vector fields on S^n?