  Brightness-temperature fluctuations in the redshifted 21-cm background from
the cosmic dark ages are generated by irregularities in the gas-density
distribution and can then be used to determine the statistical properties of
density fluctuations in the early Universe. We first derive the most general
expansion of brightness-temperature fluctuations up to second order in terms of
all the possible sources of spatial fluctuations. We then focus on the
three-point statistics and compute the angular bispectrum of
brightness-temperature fluctuations generated prior to the epoch of hydrogen
reionization. For simplicity, we neglect redshift-space distortions. We find
that low-frequency radio experiments with arcmin angular resolution can easily
detect non-Gaussianity produced by non-linear gravity with high signal-to-noise
ratio. The bispectrum thus provides a unique test of the gravitational
instability scenario for structure formation, and can be used to measure the
cosmological parameters. Detecting the signature of primordial non-Gaussianity
produced during or right after an inflationary period is more challenging but
still possible. An ideal experiment limited by cosmic variance only and with an
angular resolution of a few arcsec has the potential to detect primordial
non-Gaussianity with a non-linearity parameter of f_NL ~ 1. Additional sources
of error as weak lensing and an imperfect foreground subtraction could severely
hamper the detection of primordial non-Gaussianity which will benefit from the
use of optimal estimators combined with tomographic techniques.
