Can we express every linear transformation on $\text{M}_2(\mathbb{C})$ as sum of conjugations?

The motivation for this question comes from quantum computation, but I won't go into those details - the question seems to me as a pure linear algebra question.

Let $V=\text{M}_2(\mathbb{C})$ be the vector space of complex-valued $2\times 2$ matrices. Let $E_1,\dots ,E_n\in V$ such that $\sum E_i^*E_i = I$ and define the function $T:V\to V$ as

$T(A) \triangleq \sum E_i A E_i^*$.

1. Am I right in thinking that $T$ is a linear transformation over $V$?
2. If I am correct, my main question is: can any linear transformation over $V$ be represented in these way? i.e. for every linear transformation $T:V\to V$ one can find $E_1,\dots,E_n$ (where $n$ can depend on $T$) such that $\sum E_i^*E_i = I$ and $T(A) = \sum E_i A E_i^*$.