Computing homotopy invariant for $\pi_k \left({\mathrm{Sp}(n) \over \mathrm{U}(n)} \right)$

Question

I am focused on applications of math so please bear with me. Let $\mathrm{Sp}(n)$ be the compact symplectic group, with the isomorphism $\mathrm{Sp}(1) \cong \mathrm{SU}(2) \cong S^3$. I am interested in the form of the topological invariant for homotopy group $\pi_k\left({\mathrm{Sp}(n) \over \mathrm{U}(n)} \right)$. For $n=1$, this homotopy group is $\pi_k\left({\mathrm{Sp}(1) \over U(1)}\right)$. We have, however, that $\mathrm{Sp}(1) \cong S^3$ and $\mathrm{U}(1) \cong S^1$, meaning that, for $n=1$, the homotopy group simplifies to the homotopy group of spheres, $\pi_k\left(S^2\right)$. The topological invariant for $\pi_2\left(S^2\right)$ is known: it is the skyrmion number (perhaps winding number or topological charge is the more general term) expressed in terms of an integral, with the integrand being an expression in terms of a three-vector. But can we use this fact about $n=1$ to fix the form of the homotopy invariant for general $n$ to be a skyrmion number (winding number or topological charge)? If so, how can we determine which three-vector to use in the general $n$ case? Is it possible to determine the form of the homotopy invariant in terms of an integral directly from the general $n$ case rather than the $n=1$ case?