N-Dimensional Rotation

Hi.

I was wondering, is there a general method to build up a formula for
the rotation of an N-dimensional object about an N-2-dimensional
"axis"? Like how one rotates 2-D objects around 0-D points, 3-D objects
around 1-D lines, 4-D objects around 2-D planes, etc.?

What is the rotation formula for, say, a 10-dimensional object,
ortating around all possible 9-dimensional "axes"? What I want to do is
make a program that would show a 10-dimensional "cube" starting
"head-on" (ie. it would first look like a bunch of cubes nested inside
each other, although that's not what it would be), and then start
rotating through all the axes, tumbling round and round.

It seems there would be 24 different formulas, from the following
sequence:

1-D means there are 0 rotations
2-D means there is 1 rotation, around the origin, o, so 1 formula
3-D means there are 3 rotations (x, y, z) so 3 formulas
4-D means there are 6 rotations (xy, xz, xw, yz, yw, zw) so 6 formulas
5-D means there are 9 rotations (xyz, xyw, xyv, xzw, xzv, xwv, yzw,
yzv, zwv), so
9 formulas

What exactly would the formulas be, for each angle a1...a24?

.

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