How to compute cohomology of $C^0(S^n,X)$?

Question

While there are many approaches to the (singular) cohomology of free loop spaces $LX=C^0(S^1,X)$ in the literature, I can't find many results about the cohomology of the mapping spaces $C^0(S^n,X)$ for $n>1$, that do not require a deep understanding of homotopy theory (and its notations). In particular, I would like to know $H^*(C^0(S^n,S^{m});\mathbb{Q})$, where $m>n>1$, but can't find any explicit results on these cohomology groups. Can anyone help?