A version of Whitney's Theorem state that any $n-$dimensional manifold can be embedded in $\mathbb{R}^{2n+1}$ as a closed subset. Another version states that that any $n-$dimensional manifold can be embedded in $\mathbb{R}^{2n}$.
Is there an example of $n-$manifold which can be embedded in $\mathbb{R}^{2n}$ but not as closed subset? Or is it possible to extend Whitney's theorem to have an embedding as a closed subset in $\mathbb{R}^{2n}$.
The motivation for this question come from this comment I found reading Milnor's book on Characteristic classes (p. 120)
[...](Compare §4.8. According to [Whitney, 1944] every smooth $n-$manifold whose topology has a countable basis can be smoothly embedded in $\mathbb{R}^{2n}$. Presumably it can be embedded as a closed subset of $\mathbb{R}^{2n}$, although Whitney does not prove this).
Edit: In order to clarify what I'm looking for, you can consider the open Moebius strip: it can be embedded in $\mathbb{R}^{3}$ but not as a closed subset. I'm looking for a manifold for which this happen in the maximal dimension (i.e. $2n$).