A study of quadratic differential forms for 1-D systems was carried out in Willems and Trentelman (1998). Extention and study of this concept of quadratic differential forms to n-D systems is the main purpose of the paper. This extension opens the way to generalization of several concepts of 1-D systems, like that of conservative systems and dissipative systems and could throw new light into other control problems in the n-D case (especially in the behavioural framework). We collect together several properties of these quadratic differential forms for n-D systems and comment on their similarity and their differences from quadratic differential forms for 1-D systems.