This paper considers quadratic-in-the-state representations, which consist of state equations that are at most quadratic with respect to the states, as representations for a broad class of nonlinear systems. A necessary and sufficient condition is shown for existence of a quadratic-in-the-state representation that has the identical input-output map with a given nonlinear system. That condition is characterized by the algebraic structure of the observation space of the given system and is so mild that many types of nonlinear systems have a quadratic-in-the-state representation. A sufficient condition is also shown in terms of differentially algebraic functions, which is useful to check the existence of a quadratic-in-the-state representation for most of practical systems.