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The results of ultra-high vacuum low-temperature scanning-tunnelingmicroscopy (STM) and spectroscopy (STS) of atomically clean (111) surface ofthe topological insulator Bi$_2$Se$_3$ are presented. We observed several typesof new subsurface defects whose location and charge correspond to p-typeconduction of grown crystals. The sign of the thermoelectric effect alsoindicates p-type conduction. STM and STS measurements demonstrate that thechemical potential is always located inside the bulk band gap. We also observedchanges in the local density of states in the vicinity of the quintuple layersteps at the studied surface. This changes correspond either to the shift ofthe Dirac cone position or to the shift of the chemical potential near the stepedge.