  Eichler and Zagier developed a theory of Jacobi forms to understand and
extend Maass' work on the Saito-Kurokawa conjecture. Later Skoruppa introduced
skew-holomorphic Jacobi forms, which play an important role in understanding
liftings of modular forms and Jacobi forms. In this paper, we explain a
relation between holomorphic Jacobi forms and skew-holomorphic Jacobi forms in
terms of a group cohomology. More precisely, we introduce an isomorphism from
the direct sum of the space of Jacobi cusp forms on $\Gamma^J$ and the space of
skew-holomorphic Jacobi cusp forms on $\Gamma^J$ with the same half-integral
weight to the Eichler cohomology group of $\Gamma^J$ with a coefficient module
coming from polynomials.
