In seven dimensions, the cross product makes sense. Without resorting to nonvector tensors or exterior products (although they can be used to further explain), how does one perform this cross product both algebraically and geometrically. In particular, what is the analog to the right hand rule that determines the resultant vector uniquely? What are the directions of rotation (like "counterclockwise") in this space, how many are there, how are they defined, and which reduce to our two directions? How do they relate to the right hand rule (actually, this question applies in three dimensions too!)?
Why can the formal determinant of directions and components not be uaed to define an arbitrary n-dimensional cross product?
In general, the right hand rule is only defined in three dimensions. Whenever you go to seven dimensions with octernions or similar, you lose certain symmetries. Things like the Hodge star operator might be what you're looking for if you really want higher analogues. The most natural way to extend the cross product is with exterior algebra. This is indeed the language that higher dimensional geometry tends to be written in, like Stokes theorem: $$\int\limits_{\Omega}\mathrm{d}\omega=\int\limits_{\partial\Omega}\omega$$ $\mathrm{d}\omega$ means the exterior derivative. In three dimensions, you could use the example of corss product and right hand rule via curl: $$\iint\limits_{S} \! \nabla\times\vec{F} \, \mathrm{d}\vec{S}=\int\limits_{C} \! \vec{F} \cdot \mathrm{d}\vec{r}$$ Lie algebraes may also be of interest to you since they share many properties that cross product does.