I was reading about the interpretation of Stiefel Whitney classes as obstructions from Milnor-Stasheff's book and I got stuck at a step. The context is the following. Let $E \to B$ be a vector bundle of rank $n$ and let $V_k(\mathbb{R}^n)$ be the Stiefel manifold, i.e. the manifold whose points are $k$-tuples of linearly independent vectors in $\mathbb{R}^n$. The main theorem I want to understand is that the reduction mod 2 of the obstruction class $c_j(E)$ is equal to the Stiefel Whitney class $w_j(E) \in H^j(B; \mathbb{Z}/2)$.
Let $\gamma^n \to G_n(\mathbb{R}^{n+1}) \cong \mathbb{R}P^n$ be the canonical bundle over the Grassmannian of $n$-planes in $\mathbb{R}^{n+1}$ and consider the bundle $V_1 (\gamma^n) \to G_n(\mathbb{R}^{n+1})$ whose fibers are $V_1(\mathbb{R}^n)$. They use the following fact in their proof:
"The obstruction cocycle of the bundle $V_1 (\gamma^n) \to G_n(\mathbb{R}^{n+1})$ clearly assigns to the $n$-cell of $\mathbb{R}P^n \cong G_n(\mathbb{R}^{n+1})$ a generator of the cyclic group $\pi_{n-1}(V_1(\mathbb{R}^n))=\pi_{n-1}(\mathbb{R}^n - 0) = \mathbb{Z}$"
I understand why $\pi_{n-1}(V_1(\mathbb{R}^n))=\mathbb{Z}$. The n-th obstruction cocycle of the bundle $V_1 (\gamma^n) \to G_n(\mathbb{R}^{n+1}) \cong \mathbb{R}P^n $ assigns to the n-cell of $\mathbb{R}P^n$ an element of $\pi_{n-1}(V_1 (\gamma^n))$. So if we compose with the map induced by restriction to the fiber we get an element of $\pi_{n-1}(V_1(\mathbb{R}^n))$. Why is this element a generator of such group?
Thank you!