For $\mathbb{R}P^2$, I like Boy's surface, which is a particularly symmetric immersion of the projective plane into $\mathbb{R}^3$.

Also, see this Java-based model.

You can build a piecewise-linear version of one of these out of paper. If you cut out a disk-shaped window (to see inside to the triple point), what you have is a model of the Mobius band for which the boundary circle is really a round circle!

Of course, this doesn't really help with all of the rest of them $(n \ge 3)$!

For $\mathbb{R}P^2$, I like Boy's surface, which is a particularly symmetric immersion of the projective plane into $\mathbb{R}^3$.

Also, see this Java-based model.

You can build a piecewise-linear version of one of these out of paper. If you cut out a disk-shaped window (to see inside to the triple point), what you have is a model of the Mobius band for which the boundary circle is really a round circle!

Of course, this doesn't really help with all of the rest of them $(n \ge 3)$!