Let $\mathbb{H}$ be quaternion numbers. Let $G_n(\mathbb{H}^\infty)$ be the grassmannian of $n$-subspaces of $\mathbb{H}^\infty$. Then $H^*(G_n(\mathbb{H}^\infty),\mathbb{Z})=\mathbb{Z}[q_1,\cdots,q_n]$ with $q_i$ of dimension $4i$.
(1). Is $G_n(\mathbb{H}^\infty)$ a classifying space of $n$-dimensional quaternion vector bundles (i.e., fibre $\mathbb{H}^n$ and transition functions in $GL(\mathbb{H}^n)$) ?
(2). Let $\xi$ be a quaternion vector bundle over $X$ and $f: X\to G_n(\mathbb{H}^\infty)$ be the classifying map. Is $f^*(q_i)$ the $i$-th Pontriyagin class of $\xi$ or not?