It has been conjectured since the work of Lalley and Sellke (1987) that the branching Brownian motion seen from its tip (e.g. from its rightmost particle) converges to an invariant point process. Very recently, it emerged that this can be proved in several different ways (see e.g. Brunet and Derrida, 2010, Arguin et al., 2010, 2011). The structure of this extremal point process turns out to be a Poisson point process with exponential intensity in which each atom has been decorated by an independent copy of an auxiliary point process. The main goal of the present work is to give a complete description of the limit object via an explicit construction of this decoration point process. Another proof and description has been obtained independently by Arguin et al. (2011).