Given a (real or almost complex) manifold $M$, Let the 2-nd unordered configuration space be the quotient space $$ B(M,2)=(M\times M\setminus\ \Delta)/\ \mathbb{Z}_2 $$ where $$ \Delta=\{(m,m)\mid m\in M \} $$ and $\mathbb{Z}_2$ acts by reversing the coordinates order. Then $B(M,2)$ is a (real or almost complex) manifold. Suppose the characteristic classes of $TM$ is given. How to obtain the characteristic classed of $TB(M,2)$? For example,

(1). How to compute the Stiefel-Whitney class $$ w(TB(\mathbb{R}P^n,2))? $$

(2). How to compute the Stiefel-Whitney class $$ w(TB(\mathbb{C}P^n,2))? $$

(2). How to compute the Chern class $$ c(TB(\mathbb{C}P^n,2))? $$