  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 Gr\"obner 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.
