Doing specific rotations by breaking it into steps is possible. Rotations by 90 degrees through the higher dimensions is doable with some effort -- it's just coordinate swapping after all. You can make checks that you got it right. Once you have this mastered, you can compose it with rotations that don't touch the higher dimensions. Then compose again with one of these 90 degree rotations, and you have an effective rotation through the higher dimensions.

(Understanding the commutation relations for rotation helps in this breakdown, of course. If you can then go on to understanding how the infinitesimal rotations work, you've got the whole thing down.)