  We study the algebraic combinatorics of monomial degenerations of Pl\"ucker
forms which is governed by matching fields in the sense of Sturmfels and
Zelevinsky. We provide a necessary condition for a matching field to yield a
Khovanskii basis of the Pl\"ucker algebra for $3$-planes in $n$-space. When the
ideal associated to the matching field is quadratically generated this
condition is both necessary and sufficient. Finally, we describe a family of
matching fields, called $2$-block diagonal, whose ideals are quadratically
generated. These matching fields produce a new family of toric degenerations of
$\Gr(3, n)$.
