We investigated the local density of states (LDOS) of a normal metal (N) in good electrical contact with a superconductor (S) as a function of the distance $x$ to the NS interface. The sample consists of a pattern of alternate $L=1$ $\mu$m wide strips of Au and Nb made by UV lithography. We used a low temperature scanning tunneling microscope and a lock-in detection technique to record simultaneously $\mathrm{d}I/\mathrm{d}V(V,x)$ curves and the topographic profile $z(x)$ at $1.5$ K. We scanned along lines perpendicular to the strips. All the spectra show a dip near the Fermi energy, which spectral extension decreases from the superconducting gap $\Delta$ at the NS interface to zero at distances $x\gg\xi_{\mathrm{N}}$ where $\xi_{\mathrm{N}}\simeq\sqrt{\hbar D_{\mathrm{N}}/2\Delta}\simeq53$ nm is the coherence length in the normal metal. Our measurements are correctly described in the framework of the quasi-classical Green's function formalism. We numerically solved the $\mathrm{1D}$ Usadel equation and extracted a decoherence time in gold of $4$ ps. We also investigated the LDOS of small ridges of Au deposited on the top of the Nb lines. In this case, $L\leqslant \xi_{\mathrm{N}}$ and the spatial variations of the spectra depend on the exact shape of the Au ridge. However, our results are consistent with a predicted minigap related to the Thouless energy.