The discrete dipole approximation (DDA) is a well-known method for computation of the scattering of light from nonspherical particles. Here, we present a new scattering order formulation (SOF) of the DDA that allows the user to represent the scattering particle with higher flexibility than in conventional DDAs, while the computer memory required always scales as O(N). In our new SOF, the user can locate each dipole independently, or off-grid, in space, assign each dipole a unique size and a unique dipole shape as appropriate, and assign each dipole a unique magnetoelectric polarizability with no constraints. The cost of this flexibility is that the computation time is increased from O(N l o g N) to O(N 2). To compensate, our model allows the user to vary the range of dipole interaction in a unique manner. We find that, in cases in which the scatterer has at least one dimension that is sufficiently small compared with the wavelength, a relatively small number of iterations is required for convergence of the simulation, and in addition, a small dipole interaction range can be invoked to reduce the computation time to O(N) while still producing results that are sufficiently accurate.