With the 4th dimension, you can cheat a bit and use time itself. For instance, a 4-D sphere is something that starts at a point, grows to a sphere, then shrinks again.

It's not perfect. We can rotate in the normal 3D space but you don't get proper 4D rotation with this technique. Nor does this give you a good intuition about that shadow.

But it's something.

Gives you basically nothing when it comes to the 5th dimension, though, because I'm fresh out of spacetime at that point.