Suppose $V$ is a n-dimensional linear vector space. $\{s_1, s_2,..., s_n\}$ and $\{e_1, e_2,..., e_n\}$ are two sets of orthonormal basis with basis transformation matrix $U$ such that $e_i = \sum_j U_{ij}s_j$.
Now consider the $n^2$ dim vector space $V\bigotimes V$ (kronecker product) with equivalent basis sets $\{s_1s_1, s_1s_2,..., s_ns_n\}$ and $\{e_1e_1, e_1e_2,..., e_ne_n\}$. Now can we find the basis transformation matrix for this in terms of U?
Simply use bilinearity of the tensor product. Since, as you said, $e_i = \sum_j U_{ij} s_j$, we can write any basis vector of $V \otimes V$ as
$$ e_i \otimes e_k = \left( \sum_j U_{ij} s_j \right) \otimes \left( \sum_\ell U_{k\ell} s_\ell \right) = \sum_{k,\ell} U_{ij} U_{k\ell} s_j \otimes s_\ell $$