We determine a term order on the monomials in the variables
$\varx{i}{j}$, $1 \leq i < j \leq n$, such that
corresponding initial ideal of the ideal of Pfaffians of
degree $r$ of a generic $n$ by $n$ skew-symmetric matrix is
the Stanley-Reisner ideal of a join of a simplicial sphere and
a simplex. Moreover, we demonstrate that the Pfaffians of the
$2r$ by $2r$ skew-symmetric submatrices form a Gröobner
basis for the given term order. The same methods and similar
term orders as for the Pfaffians also yield squarefree initial
ideals for certain determinantal ideals. Yet, in contrast to
the case of Pfaffians, the corresponding simplicial complexes
are balls that do not decompose into a join as above.