Abstract: 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 (Gassner in SIAM J Sci Comput 35(3):A1233–A1253, 2013; Fisher and Carpenter in J Comput Phys 252:518–557, 2013; Carpenter et al. in SIAM J Sci Comput 36(5):B835–B867, 2014; Crean et al. in J Comput Phys 356:410–438, 2018; Chan et al. in Efficient entropy stable Gauss collocation methods, 2018. arXiv:1809.01178 ) or volume and surface quadrature rules (Chan in J Comput Phys 362:346–374, 2018; Chan and Wilcox in J Comput Phys 378:366–393, 2019) to produce operators which satisfy a summation-by-parts (SBP) property. In this work, we show how to construct “modal” DG formulations which are entropy stable for volume and surface quadratures under which the SBP property in Chan (2018) does not hold. These formulations rely on an alternative skew-symmetric construction of operators which automatically satisfy the SBP property. Entropy stability then follows for choices of volume and surface quadrature which satisfy sufficient accuracy conditions. The accuracy of these new SBP operators depends on a separate set of conditions on quadrature accuracy, with design order accuracy recovered under the usual assumptions of degree $$2N-1$$ volume quadratures and degree 2N surface quadratures. We conclude with numerical experiments verifying the accuracy and stability of the proposed formulations, and discuss an application of these formulations for entropy stable DG schemes on mixed quadrilateral-triangle meshes.