In section 12.9 from Algebraic Geometry - A First Course by Joe Harris, the author computes the dimension of the parameter space of twisted cubics, even if he doesn't give a construction of this space. How does this spaces look like? Is the construction too complicated for the level he assumes?
This computation was asked at some exam where also wasn't assumed any explicit construction. Though, at that exam it was also asked if this space is irreducible. How can one see that without knowing how the space looks like?