We exploit the potential of the spin noise spectroscopy (SNS) for studies of nuclear spin dynamics in n-GaAs. The SNS experiments were performed on bulk n-type GaAs layers embedded into a high-finesse microcavity at negative detuning. In our experiments, nuclear spin polarisation initially prepared by optical pumping is monitored in real time via a shift of the peak position in the electron spin noise spectrum. We demonstrate that this shift is a direct measure of the Overhauser field acting on the electron spin. The dynamics of nuclear spin is shown to be strongly dependent on the electron concentration.