We consider conditions for the existence of boundary modes in non-Hermitian systems with edges of arbitrary codimension. Through a universal formulation of formation criteria for boundary modes in terms of local Green's functions, we outline a generic perspective on the appearance of such modes and generate corresponding dispersion relations. In the process, we explain the skin effect in both topological and nontopological systems, exhaustively generalizing bulk-boundary correspondence to different types of non-Hermitian gap conditions, a prominent distinguishing feature of such systems. Indeed, we expose a direct relation between the presence of a point gap invariant and the appearance of skin modes when this gap is trivialized by an edge. This correspondence is established via a doubled Green's function, inspired by doubled Hamiltonian methods used to classify Floquet and, more recently, non-Hermitian topological phases. Our work constitutes a general tool, as well as a unifying perspective for this rapidly evolving field. Indeed, as a concrete application we find that our method can expose novel non-Hermitian topological regimes beyond the reach of previous methods.