I can visualize the construction of $\mathrm{RP}^2$ from a disc $B^2$ whose boundary $S^1$ is subjected to the antipodal identification. This can be obtained by glueing the edge of a Möbius strip $M$, which is $S^1$, with the edge of the disc, after suitably deforming the disc to the shape of a punctured sphere. Thus,
$$ \mathrm{RP}^2 = M \cup_{\circ} B^2$$
where $\cup_{\circ}$ denotes this glueing mechanism.
On the other hand, however, we know that $S^1$ subjected to the antipodal identification is $\mathrm{RP}^1 = S^1$. So there must be some mechanism $\cup_{?}$ such that $\mathrm{RP^{2}} = \mathrm{RP}^1 \cup_{?} B^2$. Or, more generally,
$$ \mathrm{RP^{n+1}} = \mathrm{RP}^n \cup_{?} B^{n+1} \,.$$
Does there exist any such construction? Alternatively, are there higher dimensional equivalents of Möbius strips $M^n$ such that $$ \mathrm{RP^n} = M^n \cup_{\circ} B^n \,? $$
P.S. I am not familiar with algebraic topology. So I would be most obliged if you explain in lay terms with respect to algebraic topology, should the need arise. My focus is on a visual understanding.
