Can anyone provide me with an example of an orientable closed manifold $M$ of dimension $n\geq 2$, which cannot be embedded in $\mathbb R^{2n-1}$?

I know these cannot exist for $n=1$, i.e. $S^1$. If we ignore orientability, then if we take $n=2^r$, $\mathbb RP^n$ cannot be embedded in $\mathbb R^{2n-1}$.

While searching on the internet, I found some sharper results here.

So can we say anything when $\dim(M)= 2^r$ and $M$ is orientable?

It would be very helpful if someone could provide me with some more references. Thanks in advance.