The set of lines passing through the origin in $R^n$. From this you can see that it's equivalent to the unit sphere modulo reflections. Or the upper half of the unit sphere with the antipodal points on the boundary identified with each other.

Also, you can parameterize a dense open set using the hyperplane $x^n = 1$; the only lines you miss are the ones parallel to this hyperplane, i.e., the lines lying inside $x^n = 0$. Obviously, you can do this with each co-ordinate, giving you $n$ dense open sets, each with a natural map to $R^{n-1}$ (called "affine co-ordinates"), that cover $RP^{n-1}$.

I assume that these descriptions don't do it for you. Can you say more about what you want?

The set of lines passing through the origin in $R^n$. From this you can see that it's equivalent to the unit sphere modulo reflections. Or the upper half of the unit sphere with the antipodal points on the boundary identified with each other.

Also, you can parameterize a dense open set using the hyperplane $x^n = 1$; the only lines you miss are the ones parallel to this hyperplane, i.e., the lines through the origin lying inside $x^n = 0$. The set of missing lines is therefore itself an $RP^{n-2}$. So $RP^{n-1}$ can be viewed as the union of $R^{n-1}$ with $RP^{n-2}$ (which is usually called the "hyperplane at infinity").

Obviously, you can do this with each co-ordinate, giving you $n$ dense open sets, each with a natural map to $R^{n-1}$ (called "affine co-ordinates"), that cover $RP^{n-1}$.

I assume that these descriptions don't do it for you. Can you say more about what you want?