De Rham cohomology of $T^n$ using Künneth formula and Chevalley-Eilenberg theorem.

Question

I want to calculate $H^*(T^n)$ with ring structure using both of these methods. Künneth formula gives $$ H^p(T^n)=H^p(S^1\times T^{n-1})=\bigoplus_{i+j=p}H^i(S^1)\otimes H^j(T^{n-1}) $$ for each $p$,but $H^j(S^1)$ is only nonzero for $j=0$ and $j=1$, so I get $$ H^p(T^n)=\mathbb{R}\otimes H^p(T^{n-1})\oplus \mathbb{R}\otimes H^{p-1}(T^{n-1}) $$ From now, it should be more or less clear that proceeding by induction I am just going to get $\mathbb{R}^{\binom{n}{p}}$, but how do I obtain the ring structure for that? Using Chevalley-Eilenberg, I know that $$ H^*(K_{T^n})\cong H^*(T^n), $$ but I don't know how to actually get the Lie algebra cohomology. How do I proceed in both cases and show that I get isomorphic answers?