So first for n even, $RP^n$ is not orientable, hence can not be embedded in $\mathbb{R}^{n+1}$.

For odd n, $RP^{n}$ is orientable, hence the normal bundle is trivial. Now using stiefel-Whitney classes, one can prove when $n$ not of the form $2^k - 1$, it can not be embedded in $\mathbb{R}^{n+1}$. I would appreciate if someone can give an more elementary proof in this case.

Then for the left cases. $RP^3$ can not be embedded proving by homology thoery (Alexander sphere duality, lefschetz duality and a long exact sequence).

For the other cases, I do not know how to prove. I realized Don Davis has a table for the immersion and embedding of $RP^n$ (http://www.lehigh.edu/~dmd1/immtable). But the question I am asking is easier, hence there may be an answer and a proof I could follow.