Let $Sp(n)=U(n,\mathbb{H})=\{A \in M_{n}(\mathbb{H}) : A\cdot A^{*}=I\}$ be the compact symplectic group, a subset of $Sp(2n,\mathbb{C})$. I want to show that $Sp(n)$ is simply connected, in accordance with this Wikipedia article.
I know that for the case $n=1$, we can write elements of $Sp(1)$ in the form $\cos\theta+v\sin\theta$, where $v$ is a purely imaginary unit quaternion; this gives us a nice, explicit path to the identity. I'm not sure what argument to make for higher dimensions, though. I don't have much experience in dealing with Lie groups or smooth manifolds, so I'm not familiar with any "standard" methods of proving simple connectedness in these objects.
Thanks for any help you can offer.