How Do You Calculate Surface Integrals in Higher Dimensions?

[logarithmic]

How would I calculate a surface integral in dimensions greater than 3.

For example, from the definition of a surfrace integral over a vector field: http://en.wikipedia.org/wiki/Surface_integral#Surface_integrals_of_vector_fields

To compute the surface integral, I would first need a vector normal to the vector field. In R^3 this is just done by using the cross product. Is there a general way to find a normal vector when not in R^3, since the cross product is no longer valid?

 

[Hurkyl]

To compute the surface integral, I would first need a vector normal to the vector field.

Actually, what you need is the a bivector tangent to the surface.

That trick works in R³ because bivectors can be identified with "axial vectors". However, in R⁴, the dual would be some sort of "axial bivector", and in R⁵ it would be an "axial trivector" -- so we can't use this trick anymore.