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 $(x_i x_j)/|x|^2$.
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.