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Variational space-time formulations for partial differential equations have been of great interest in the last decades, among other things, because they allow to develop mesh-adaptive algorithms. Since it is known that implicit time marching schemes have variational structure, they are often employed for adaptivity. Previously, Galerkin formulations of explicit methods were introduced for ordinary differential equations employing specific inexact quadrature rules. In this work, we prove that the explicit Runge-Kutta methods can be expressed as discontinuous-in-time Petrov-Galerkin methods for the linear diffusion equation. We systematically build trial and test functions that, after exact integration in time, lead to one, two, and general stage explicit Runge-Kutta methods. This approach enables us to reproduce the existing time-domain (goal-oriented) adaptive algorithms using explicit methods in time. • A variational formulation of explicit Runge-Kutta methods is derived. • Exact integration, instead of inexact quadrature rules, is employed. • A systematic construction of the trial and test functions is proposed for any stage. • The formulation is applied to goal-oriented error estimation. [ABSTRACT FROM AUTHOR]