The Monodromy and Parallel Form

Question

According to famous Riemann-Hilbert correspondence, a flat connection gives a representation of fundamental group which is called monodromy. I would like to ask how does the monodromy relate to special form on the vector bundle.Here is my question:
1.Assume the $C^{r}\rightarrow E\rightarrow M$ is a vector bundle with flat connection $\nabla$, if the monodromy $\rho^{\nabla}:\pi_{1}(M,p)\rightarrow SL(r,C)$, show that: the $det$ is a parallel form on $\wedge^{r}E$.
2.If the monodromy lies in $U(r)$, then the vector bundle admits a parallel hermitian metric.
This is my idea: I try to directly compute it, the question means $\nabla det=0$, then I choose any frame $s_{1},...,s_{r}$, we have $\nabla det(s_{1},...,s_{}r)=d(set(s_{1},...,s_{r}))-\sum det(s_{1},...,\nabla s_{i},...,s_{r}).$ But how to show this is identically zero? I know because of flatness, we can locally choose parallel frame, but I think this doesn't give any help after my computation. Could you give me some help?