In my reference, Page 365, Operator-sum representation, Chapter 8, Quantum Computation and Quantum Information by Nielsen and Chuang, it is given that
We have a principal system $Q$ and an environment $E$ and $U$ is a unitary operator acting on the combined system $QE$. \begin{align} \mathcal{E}(\rho)&=tr_E\bigg(U(\rho\otimes|e_0\rangle\langle e_0|)U^\dagger\bigg)\\ &=\sum_k(I\otimes\langle e_k|)(U(\rho\otimes|e_0\rangle\langle e_0|)U^\dagger)(I\otimes |e_k\rangle)\\ &=\sum_k\langle e_k|(U(\rho\otimes|e_0\rangle\langle e_0|)U^\dagger)|e_k\rangle\\ &=\sum_k(I\otimes\langle e_k|)U(I\otimes|e_0\rangle)\rho(I\otimes\langle e_0|)U^\dagger(I\otimes|e_k\rangle)\\ &=\sum_k\langle e_k|U|e_0\rangle\rho\langle e_0|U^\dagger|e_k\rangle\\ &=\sum_k E_k \rho E_k^\dagger \end{align} where $E_k=(I\otimes\langle e_k|)U(I\otimes|e_0\rangle)=\langle e_k|U|e_0\rangle$ is an operator on the state space of the principal system $Q$, $|e_k\rangle$ are the orthonormal basis vectors of the environment, $|e_0\rangle$ be the initial state (some standard state) of the environment.
Please check Prove that $tr\Big(\sum_k E_k^\dagger E_k\rho\Big)=1$ for all $\rho$ implies $\sum_k E_k^\dagger E_k=I$ for the derivation.
In the problem, $E_k=(I\otimes\langle e_k|)U(I\otimes|e_0\rangle)=\langle e_k|U|e_0\rangle$ where $|e_k\rangle$ are some orthonormal basis vectors and $|e_0\rangle$ be some standard state of the environment $E$.
How do we say that $U$ is represented as the block matrix $$ U=\begin{bmatrix} [E_1] & \cdots & \cdots\\ [E_2] & \cdots & \cdots\\ [E_3] & \cdots & \cdots\\ \vdots & \vdots & \vdots \end{bmatrix} $$ in the basis $|e_k\rangle$ ?
My observation:
If we had an operator $W:V\to V$ and $\{|v_i\rangle\}$ constitute a basis of $V$ then it can be shown that $\langle v_i|W|v_j\rangle$ is the $ij^{th}$ term of the operator $W$, i.e., $$ W=\begin{bmatrix}\langle v_1|W|v_1\rangle&\langle v_1|W|v_2\rangle&\cdots\\ \langle v_2|W|v_1\rangle&\langle v_2|W|v_2\rangle&\cdots\\ \vdots&\vdots&\vdots \end{bmatrix} $$ is the representation of the operator $W$ in the basis $\{|v_i\rangle\}$. This is clear!
Unlike $\langle v_i|W|v_j\rangle$ which is a number and $|v_j\rangle$ is a vector, the term $E_k=(I\otimes\langle e_k|)U(I\otimes|e_0\rangle)=\langle e_k|U|e_0\rangle$ is an operator acting on the first system (principal system $Q$) and $I\otimes|e_0\rangle=|e_0\rangle,I\otimes|e_k\rangle=|e_k\rangle$ are matrices.