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.