Let $\tau:=(T\mathbb{R}P^n, \pi, \mathbb{R}P^n)$ be the tangent bundle of $\mathbb{R}P^n$. The group $GL(n,\mathbb{R})$ acts on $\pi^{-1}(x) \simeq \mathbb{R}^n$ so we can consider $T\mathbb{R}P^n$ as a $GL(n)$-principal bundle and we can reduce the structure group to $O(n)$ using Gramm-Schmidt. Now we can consider the classificant map $$f: \mathbb{R}P^n \rightarrow BGL(n) \simeq Gr_{n}(\mathbb{R}^{\infty})$$ and the reduced map $$\tilde{f}: \mathbb{R}P^n \rightarrow BO(n) \simeq Gr_{n}(\mathbb{R}^{\infty})$$ How can write explicitely this maps? We know that $\tau$ isn't a principal bundle, but why we can study the associated bundle $GL(\mathbb{R}P^n)$ in order to understand the orientation of $\mathbb{R}P^n$?
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If you want an explicit construction of the classifying map you can consider the embedding of $\mathbb RP^n$ into $\mathbb R^{(n+1)^2}$ which sends a line to the matrix representing the orthogonal projection onto that line. – Alexander Thumm May 23 '13 at 18:31