The results of ultra-high vacuum low-temperature scanning-tunneling microscopy (STM) and spectroscopy (STS) of atomically clean (111) surface of the topological insulator Bi$_2$Se$_3$ are presented. We observed several types of new subsurface defects whose location and charge correspond to p-type conduction of grown crystals. The sign of the thermoelectric effect also indicates p-type conduction. STM and STS measurements demonstrate that the chemical potential is always located inside the bulk band gap. We also observed changes in the local density of states in the vicinity of the quintuple layer steps at the studied surface. This changes correspond either to the shift of the Dirac cone position or to the shift of the chemical potential near the step edge. ; Comment: 4 pages, 8 figures; missed page inserted