An experimental investigation of a homogeneous swarm of rising bubbles is presented. The experimental arrangement ensures that all the bubbles have the same equivalent radius, a = 1.25 mm. This particular size corresponds to high-Reynolds-number ellipsoidal rising bubbles. The gas volume fractions α is small, ranging from 0.5 to 1.05%. The results are compared with the reference situation of a single rising bubble, which was investigated in a previous work. From the use of conditional statistics, the existence of two regions in which the liquid velocity fluctuations are of a different nature are distinguished. In the vicinity of the bubbles, the liquid fluctuations are the same as those measured close to a single rising bubble. They therefore do not depend on α. Far from the bubble, the liquid fluctuations are controlled by the nonlinear interactions between the wakes of all the bubbles. Their probability density function scales as α0.4, exhibiting a self-similar behaviour. The total fluctuation combines the contributions of these two regions weighted by the fraction of the liquid volume they occupy. The contribution of the bubble vicinity is thus shown to vary linearly with α while the wake contribution does not. Both are non-isotropic since strong upward vertical fluctuations are more probable.