A closed smooth $n$-manifold embeds into $\mathbb R^{2n-1}$ if and only if the normal $(n-1)$th Stiefel-Whitney class vanishes. This is due to Hirsch-Haefliger in dimensions $\neq 4$ and to Fang in dimension $4$. Massey showed that if the normal $(n-1)$th Stiefel-Whitney class is nonzero, then $M$ is non-orientable and $n$ is a power of $2$. Thus any smooth closed orientable $n$-manifold smoothly embeds into $\mathbb R^{2n-1}$.
For references see <a href="http://www.sciencedirect.com/science/article/pii/0040938363900284">here</a> and <a href="http://www.sciencedirect.com/science/article/pii/0040938394900221">here</a>.