While dealing with $BO(n)$, $BSO(n)$ and $BSpin(n)$ with the universal coefficient theorem and Künneth formula, I came to have the following question:

The universal coefficient says $H^n(X;M)\cong \hom(H_{n}(X;\mathbb{Z}),M)\oplus {\rm Ext}^{1} (H_{n-1}(X;\mathbb{Z},M))$ for a $\mathbb{Z}$-module $M$.

When $X=BSpin(n)$, we know that $H^4(BSpin(n);\mathbb{Z})\cong \mathbb{Z}$ and it seems likely that once we know what $H_p(BSpin(n);\mathbb{Z})$ would be for $p=3,4$ we might be able to retrieve this isomorphism with the aid of universal coefficient theorem.

So what would be $H_p(BSpin(n);\mathbb{Z})$ at least for $p=0,1,2,3,4$?

(I have to say that the question is not about how to prove the isomorphism $H^4(BSpin(4);\mathbb{Z})\cong \mathbb{Z}$.)

Thanks!