Proposition 17.6.1 of "Differential form in Algebraic Topology" by Bott and Tu proves the following beautiful result:

$[S^{q}, X]\simeq \frac{\pi_{q}(X,x)}{\pi_{1}(X,x)}$

where $S^{q}$ is the $q$-sphere, $X$ is any path connected space and $[S^{q}, X]$ is the set of homotopy classes of continuous maps from $S^{q}$ to $X$. I was wondering if there was an analogous result for the product of spheres, namely:

$[S^{q}\times S^{p}, X] = ?$

I am mostly interested in the cases where $S^{q}$ and $S^{p}$ are both trivializable and $X$ is the classifying space for $U(k)$, namely $X = BU(k)$. In other words, I am interested in the set of equivalence classes of complex vector bundles of rank $k$ on $S^{q}\times S^{p}$.

Thanks.