When you have an Euclidean vector space of type: $$\mathbb{R}^n\rightarrow M\times_{S_n}\mathbb{R}^n\rightarrow M/S_n$$ Then its classifying map $\phi:M/S_n\rightarrow BO(n)$ factors as: $$M/S_n\rightarrow BS_n\stackrel{\rho_n}{\rightarrow} BO(n)$$ where $\rho_n$ is the induced by the regular reprsentation $S_n\rightarrow O(n)$.

Thus what you want to compute is the morphism $$\rho_n^*:H^*(BO(n),\mathbb{F}_2)\rightarrow H^*(BS_n,\mathbb{F}_2)$$ A nice way to formulate such a result is to put all $n$ together. Then you have a map of Hopf rings $$\rho:H^*(BO(\bullet),\mathbb{F}_2)\rightarrow H^*(BS_{\bullet},\mathbb{F}_2).$$ Then set $w_{i,n}=\rho^*_n(w_i)\in H^i(BS_n,\mathbb{F}_2)$ where $w_i$ is the i-th Stiefel-whitney class and $w(k,l)=w_{2^k(2^l-1),2^{k+l}}$. Then Giusti, Salvatore and Sinha proved in their paper "Mod-2 cohomology of symmetric groups as a Hopf ring" (theorem $10.7$):

"The classes $w(k,l)$ generate $H^*(BS_{\bullet},\mathbb{F}_2)$ as a Hopf ring."

When you have an Euclidean vector space of type: $$\mathbb{R}^n\rightarrow M\times_{S_n}\mathbb{R}^n\rightarrow M/S_n$$ Then its classifying map $\phi:M/S_n\rightarrow BO(n)$ factors as: $$M/S_n\rightarrow BS_n\stackrel{\rho_n}{\rightarrow} BO(n)$$ where $\rho_n$ is induced by the regular reprsentation $S_n\rightarrow O(n)$.

Thus what you want to compute is the morphism $$\rho_n^*:H^*(BO(n),\mathbb{F}_2)\rightarrow H^*(BS_n,\mathbb{F}_2)$$ A nice way to formulate such a result is to put all $n$ together. Then you have a map of Hopf rings $$\rho:H^*(BO(\bullet),\mathbb{F}_2)\rightarrow H^*(BS_{\bullet},\mathbb{F}_2).$$ Then set $w_{i,n}=\rho^*_n(w_i)\in H^i(BS_n,\mathbb{F}_2)$ where $w_i$ is the i-th Stiefel-whitney class and $w(k,l)=w_{2^k(2^l-1),2^{k+l}}$. Then Giusti, Salvatore and Sinha proved in their paper "Mod-2 cohomology of symmetric groups as a Hopf ring" (theorem $10.7$):

"The classes $w(k,l)$ generate $H^*(BS_{\bullet},\mathbb{F}_2)$ as a Hopf ring."