Dual group of the torus

Question

Let $T^n=(S^1)^n$ be the torus. We can consider the dual group $\widehat{T^n}=Hom(T^n, \Bbb{C}^*)=\{\phi:T^n\to \Bbb{C}^* : \phi \text{ a group homomorphism}\}$ then $\widehat{T^n}\cong \Bbb{Z}^n$. How does one prove this? I can see for every integer vector $(a_1, \dots, a_n)$ the map $(t_1, \dots, t_n)\mapsto t_1^{a_1}\cdots t_n^{a_n}$ is a homomorphism. But how to show this is the only type of homomorphism? This seems to be a basic fact but I am unable to figure it out. Thanks for the help.