Let $S^m$ be the $m$-sphere and $$F(S^m,2)/\mathbb{Z}_2=\{(a,b)\mid a,b\in S^m, a\neq b\}/(a,b)\sim (b,a)$$ be the $2$-nd unordered configuration space on $S^m$. How to compute the total Stiefel-Whitney class of the tangent bundle of $F(S^m,2)/\mathbb{Z}_2$ $$ w(TF(S^m,2)/\mathbb{Z}_2)? $$

Is $F(S^m,2)/\mathbb{Z}_2$ homeomorphic to $\mathbb{R}P^m\times \mathbb{R}^m$ or not?

Moreover, if we change $S^m$ to other manifolds, for example, projective spaces, Grassmannians, then how to compute the corresponding Stiefel-Whitney class? The question is on characteristic classes of tangent bundle of 2-nd unordered configuration space