Probably we can't. But why?
My reasoning is that any topological immersion of $\mathbb{P}^2$ in $\mathbb{R}^3$ (e.g. cross-cap, Roman surface) seems to intersect itself (I will find a proof or disproof of it later, but now suppose that's true), and so for a point $p$ on the intersection, among open sets defined by Euclidean metric, we can't find an (open) neighborhood of $p$. Therefore with Euclidean metric we can't make a topological immersion of $\mathbb{P}^2$ in $\mathbb{R}^3$ a manifold.
Is that reasoning correct?