  In this paper we provide an analytical procedure which leads to a system of
  $(n-2)^2$ polynomial equations whose solutions give the parameterisation of
the complex $n\times n$ Hadamard matrices. It is shown that in general the
Hadamard matrices depend on a number of arbitrary phases and a lower bound for
this number is given. The moduli equations define interesting geometrical
objects whose study will shed light on the parameterisation of Hadamard
matrices, as well as on some interesting geometrical varieties defined by them.
