Given a topological group $G$ and a subset $S$ of $G$ that topologically generates it, what are the conditions under which an $n$-dimensional continuous linear representation of $G$ over an algebraically closed topological field $k$ can be constructed once we have specified a collection of degree $n$ polynomials with coefficients in $k$ as the characteristic polynomials of the images of elements of $S$?

In other words, when can a continuous homomorphism $\rho: G \rightarrow GL_n(k)$ be constructed from the data $\{c(\rho (s)) \in k[x]: s \in S\}$ where $c(a)$ is the characteristic polynomial of $a$?

Under which further conditions if any is such a representation uniquely determined? Is there an algorithm for constructing it from the collection of characteristic polynomials?

Finally what if we don't require that $k$ be algebraically closed?

Assuming that $k$ is a field of characteristic 0 would be fine, although I am also interested in the characteristic $p$ case.