One way is to relax right angles. I.e., imagine 4d space without orthogonal bases.
Take a pyramid. The top point is the origin. Each of the edges that run down the sides are axes. Let's call them x-, y-, z- and w-.
Imagine a line from the origin to the middle of the square base at the bottom. That is the x = y = z = w line.
A pentagonal pyramid would suffice for 5d. And interestingly enough, a cone would suffice for infinite-d. It's sometimes helpful to think that each of these axes 'act' on the line or planar 'shape' (the, x = y = z = w line is perhaps one of the simplest, but of course any equation can be visualized within the pyramid or cone...).
One other thing to note, however, is that this is just one 'view' or projection of 4d space into 3d space. Real orthogonal 4d geometric space isn't (I don't think) viewable in orthogonal 3d geometric space. We can only see a projection of what it looks like.