Question regarding the fundamental group of $SU(n)$

Question

I want to prove that $\pi_1 (GL(n,\mathbb{C})) \simeq \mathbb{Z}$. To do so, i alredy proved that $GL(n,\mathbb{C})$ and $U(n)$ have same homotopy type and that there is and homeomorphism between $U(n)$ and $SU(n) \times S^1$. But i'm a bit lost in proving that $SU(n)$ is simply connected. I know that there is a proof using fibrations and long exact sequences of homotopy groups, but i don't know much about either yet. My current knowledge is fundamental groups, the Van Kampen Theorem and covering spaces.