  Based on the realization of three-algebras in terms of algebra of matrices
and four-brackets [arXiv:0807.1570] we present the notion of u(N)-based
extended three-algebras, which for N=2 reproduces the Bagger-Lambert
three-algebra. Using these extended three-algebras we construct an su(N)\times
su(N) Chern-Simons action with explicit SO(8) invariance. The dynamical fields
of this theory are eight complex valued bosonic and fermionic fields in the
bi-fundamental representation of the su(N)\times su(N). For generic N the
fermionic transformations, however, close only on a subclass of the states of
this theory onto the 3d, {\cal N}=6 superalgebra. In this sector we deal with
four complex valued scalars and fermions, our theory is closely related to the
ABJM model [arXiv:0806.1218], and hence it can be viewed as the (low energy
effective) theory of N M2-branes. We discuss that our three-algebra structure
suggests a picture of open M2-brane stretched between any two pairs of
M2-branes. We also analyze the BPS configurations of our model.
