Expanding on the comment by @user127776, the key reference is Palais, "Lusternik-Schnirelman Theory on Banach Manifolds", Topology 5 (1966), where it is proved that if $X$ can be covered by $n$ contractible closed sets, then the cup-length of $X$ is strictly less than $n$.
(Here the cup-length is the largest $n$ such that for some field $F$ and some elements $c_1,\ldots,c_n$ in $H^*(X,F)$, we have $c_1\cup\ldots\cup c_n\neq 0$.)
This rules out covering ${\mathbb RP}^n$ with $n$ closed contractible sets, which should suffice here (after slightly shrinking the given $n$ copies of ${\mathbb R}^n$).