Context:
I am trying to show that $H^{\ast}(SU(n))\cong\Lambda[x_{3},\dots x_{2n-1}]$.
My approach was to use induction- $SU(2)\cong S^3$ so $H^{\ast}(SU(2))\cong H^{\ast}(S^3)\cong\Lambda[x_{3}]$ so the base case is easy. Then, I tried to apply the Serre spectral sequence on $$SU(n)\to SU(n+1)\to S^{2n+1}$$
assuming that $H^{\ast}(SU(n))\cong\Lambda[x_{3},\dots x_{2n-1}]$. The $E_{2}$ page will look like:
$$E_{2}^{p,q}=H^{p}(S^{2n+1};H^{q}(SU(n))=\begin{cases}H^{q}(SU(n)) & \text{ if } p=0,2n+1 \\ 0 & \text{ otherwise } \end{cases}$$
However, I realised that I wasn't sure if I could deduce what $H^{p}(SU(n))$ was from $H^{\ast}(SU(n))$.
So my question is: can I deduce $H^{p}(SU(n))$ from $H^{\ast}(SU(n))$? If so, how and what will it be?