By thermally characterizing nanometer-thin suspended silicon membranes with various micrometric lengths in ambient conditions, we determine simultaneously the spatial resolution of our Wollaston-probe scanning thermal microscopy experiment, which probes an area of (285 nm)², and the effective thermal conductivity of the membranes of 40 W.m^-1.K^-1. This value is smaller thanthe in-plane thermal conductivity measured using other techniques in vacuum (60 W.m^-1.K^-1), revealing that both cross-plane and in-plane heat conduction are strongly affected by the native oxide in ambient conditions. This work also underlines that high-thermal conductivity samples can be characterized by scanning thermal microscopy when micro-patterned.