We consider the problem of state observation for systems having a well-defined observability canonical form (Gauthier and Kupka (2004)) by means of high-gain observers. The main goal is to show that, for this class of systems, observers can be designed with the high-gain parameter powered just up to the order 2 regardless the dimension of the state system. In this way we substantially overtake the main limitations of standard design procedures in which the high-gain parameter is powered up to the order of the system. The observer structure, which generalizes the ideas presented in Astolfi and Marconi (2015), can be used in all those contexts where fast state observation is required, such as in the design of output feedback stabilizers by means of the nonlinear separation principle that is also specifically addressed in the paper.