Path integral formulation for Green's functions

In the first place, I am struggling when trying to derive the path integral formulation of the Green function for non-interacting particles

$$G_{ij}(tau)=-frac{1}{Z}int D(bar{psi},psi) psi_i(tau)bar{psi}_j(0)e^{-iS(bar{psi},psi)}$$

with the action $$S=sum_iint^{beta}_{0}dtau bar{psi}_i(tau)(ipartial_{tau}+epsilon_i-mu)psi_i(tau),$$ from the definition

$$G_{ij}(tau)=-langle text{T}_tau psi_i(tau)psi_j^dagger(0)rangle$$

with $text{T}_tau$ being the time-ordering operator and $psi_i(tau)=e^{hat{H}tau}psi_i(0)e^{-hat{H}tau}$. I am told to use the fact that $langle hat{A} rangle =-frac{1}{Z}text{tr}(e^{-betahat{H}}hat{A})$ and I know the general construction allowing to express traces in terms of path integrals, but the latter requires the operators in the argument of the trace to be normal ordered. If I try to rearrenge them in order to achieve this, the calculations get really messy. Do you have any ideas on what I am missing?

Secondly, how can I rewrite the path integral formulation of $G_{ij}(tau)$ as a Matsubara summation via Gaussian integrals? In fact, the integral in the exponent prevents me to have a prototypical Gaussian integral which I can rewrite right away. It seems I am missing something here as well.