  We prove that for any open Riemann surface $N,$ natural number $n\geq 3,$
non-constant harmonic map $h:N\to \mathbb{R}^{n-2}$ and holomorphic 2-form $H$
on $N,$ there exists a weakly complete harmonic map $X=(X_j)_{j=1,\ldots,n}:N
\to \mathbb{R}^n$ with Hopf differential $H$ and $(X_j)_{j=3,\ldots,n}=h.$ In
particular, there exists a complete conformal minimal immersion
$Y=(Y_j)_{j=1,\ldots,n}:N \to \mathbb{R}^n$ such that $(Y_j)_{j=3,\ldots,n}=h.$
  As a consequence of these results, complete full non-decomposable minimal
surfaces with arbitrary conformal structure and whose generalized Gauss map is
non-degenerate and fails to intersect $n$ hyperplanes of $\mathbb{CP}^{n-1}$ in
general position are constructed. Moreover, complete non-proper embedded
minimal surfaces in $\mathbb{R}^n,$ $\forall n>3,$ are exhibited.
