Cross products likely won't vanish, precisely because they're heavily linked to the Clifford algebra, and are essentially all you need for the most common case.

By the way, it is possible to generalize the cross product to higher dimensions, it just won't be a nice binary product, in n dimensions you need n-1 vectors to uniquely determine one orthogonal to all of them. Of course you can achieve pretty much the same thing with the Clifford algebra, or with wedge products.