You might enjoy Problem 1B from Milnor and Stasheff's book Characteristic Classes. The problem produces an explicit (if rather high-dimensional) embedding of $RP^n$ into Euclidean space. Specifically, it turns out that $RP^n$ can be viewed as the space of $(n+1)\times(n+1)$ real symmetric projection matrices with trace 1. This may not be so helpful for visualization, but it is quite concrete.

You might enjoy Problem 1B from Milnor and Stasheff's book Characteristic Classes. The problem produces an explicit (if rather high-dimensional) embedding of $RP^n$ into Euclidean space. Specifically, it turns out that $RP^n$ can be viewed as the space of $(n+1)\times(n+1)$ real symmetric projection matrices with trace 1. This may not be so helpful for visualization, but it is quite concrete.

This embedding comes from the map sending a vector $x = (x_1, \ldots, x_{n+1})\in \mathbb{R}^{n+1}$ to the matrix with $(i,j)^{\textrm{th}}$ entry $(x_i x_j)/|x|^2$.