  We define a 3-algebra with structure constants being symmetric in the first
two indices. We also introduce an invariant anti-symmetric tensor into this
3-algebra and call it a symplectic 3-algebra. The general N=5 superconformal
Chern-Simons-matter (CSM) theory with SO(5) R-symmetry in three dimensions is
constructed by using this algebraic structure. We demonstrate that the
supersymmetry can be enhanced to N=6 if the sympelctic 3-algebra and the fields
are decomposed in a proper fashion. By specifying the 3-brackets, some
presently known N=5, 6 superconformal theories are described in terms of this
unified 3-algebraic framework. These include the N=5, Sp(2N) X O(M) CSM theory
with SO(5) R-symmetry , the N=6, Sp(2N) X U(1) CSM theory with SU(4)
R-symmetry, as well as the ABJM theory as a special case of U(M) X U(N) theory
with SU(4) R-symmetry.
