  We extend to the multidimensional case a Wong-Zakai-type theorem proved by Hu
and {\O}ksendal in [7] for scalar quasi-linear It\^o stochastic differential
equations (SDEs). More precisely, with the aim of approximating the solution of
a quasilinear system of It\^o's SDEs, we consider for any finite partition of
the time interval $[0,T]$ a system of differential equations, where the
multidimensional Brownian motion is replaced by its polygonal approximation and
the product between diffusion coefficients and smoothed white noise is
interpreted as a Wick product. We remark that in the one dimensional case this
type of equations can be reduced, by means of a transformation related to the
method of characteristics, to the study of a random ordinary differential
equation. Here, instead, one is naturally lead to the investigation of a
semilinear hyperbolic system of partial differential equations that we utilize
for constructing a solution of the Wong-Zakai approximated systems. We show
that the law of each element of the approximating sequence solves in the sense
of distribution a Fokker-Planck equation and that the sequence converges to the
solution of the It\^o equation, as the mesh of the partition tends to zero.
