  For each partition of the positive integer $n= \ell +\sum_{j=1}^\ell d_j$,
where $\ell\ge 1$ and $d_j \ge 0$ are integers, we construct a continuous
$(\ell-1)$-parameter family of explicit complete gradient steady
K\"ahler--Ricci solitons on $\mathbb{C}^n$ admitting a hamiltonian $2$-form of
order $\ell$ and symmetry group ${\rm U}(d_1+ 1) \times \cdots \times {\rm
U}(d_{\ell} + 1)$. For $\ell=1, \, d_1=n-1, \, d_2=\cdots = d_{\ell}=0$ we
obtain Cao's example [17] whereas for other partitions the metrics are new.
Furthermore, when $n=2, \, \ell=2, \, d_1=d_2=0$ we obtain complete gradient
steady K\"ahler--Ricci solitons on $\mathbb{C}^2$ which have positive sectional
curvature but are not isometric to Cao's ${\rm U}(2)$-invariant example. This
disproves a conjecture by Cao. We also present a construction yielding explicit
families of complete gradient steady K\"ahler-Ricci solitons on $\mathbb{C}^n$
containing higher dimensional extensions of the Taub-NUT Ricci-flat K\"ahler
metric on $\mathbb{C}^2$. When $n\ge 3$, the complete Ricci-flat K\"ahler
metrics, and when $n\ge 2$, their deformations to complete gradient steady
K\"ahler Ricci solitons seem not to have been observed before our work.
