The holomorphic version of Galois theory

Say $a_n=1$. You can obtain this map as a section of the map $\sigma$ sending the $n$-tuple of the roots $(r_1,\ldots,r_n)$ to the coefficients of the polynomial using the symmetric polynomials, corresponding to the equality $$\prod _{j=1}^n (z-r_j) = \sum_{j=0}^n a_j z^j$$

This map is holomorphic, locally biholomorphic outside the union $\Delta$ of the diagonals ${r_j=r_i}$, $i\neq j$ (corresponding to multiple roots). $\sigma$ is a holomorphic covering of degree $n!$ outside $\Delta$, and the Riemann manifold of its "inverse" exists and provides a manifold $\hat M$ which can be compactified as a Riemann manifold $M$ for "$\sigma^{-1}$" (since $\sigma$ is polynomial).

The topological structure of the covering space $\hat M$ is that of the complement of the hyperplanes arrangement given by $\Delta$, so its fundamental group will be a braid group.

As you mentioned what you can write down is limited by Galois theory, so you're not going to have anything "explicit" starting from degree 5, so I'm afraid you'll have to be satisfied with the above "inverse" description.

The thing you are asking was much studied in connection with Hilbert Problem 13. The roots of a polynomial of degree exactly $d$ form an unordered $d$-tuple. The set of unordered $d$-tuples is called the configuration space. It is the factor of $C^d$ over the action of permutation group. It is equivalent to the space of polynomials of degree exactly $d$ modulo multiplication by a non-zero constant. One recent reference is http://arxiv.org/pdf/math/0403120v3, and it contains many other references. The version of Hilbert problem 13 asks whether this function, mapping a polynomial to its roots, can be represented as a composition of functions of fewer variables.