  High order entropy stable discontinuous Galerkin (DG) methods for nonlinear
conservation laws satisfy an inherent discrete entropy inequality. The
construction of such schemes has relied on the use of carefully chosen nodal
points or volume and surface quadrature rules to produce operators which
satisfy a summation-by-parts (SBP) property. In this work, we show how to
construct "modal" skew-symmetric DG formulations which are entropy stable for
volume and surface quadratures under which a traditional SBP property does not
hold. These skew-symmetric formulations avoid the use of a "strong"
matrix-based SBP property, and instead rely on the assumption that discrete
operators exactly differentiate constants and satisfy a discrete form of the
fundamental theorem of calculus. We conclude with numerical experiments
verifying the accuracy and stability of the proposed formulations, and discuss
an application of skew-symmetric formulations for entropy stable DG schemes on
mixed quadrilateral-triangle meshes.
