  We introduce a generalization, called a skew Clifford algebra, of a Clifford
algebra, and relate these new algebras to the notion of graded skew Clifford
algebra that was defined in 2010. In particular, we examine homogenizations of
skew Clifford algebras, and determine which skew Clifford algebras can be
homogenized to create Artin-Schelter regular algebras. Just as (classical)
Clifford algebras are the Poincar\' e-Birkhoff-Witt (PBW) deformations of
exterior algebras, skew Clifford algebras are the $\mathbb{Z}_2$-graded PBW
deformations of quantum exterior algebras. We also determine the possible
dimensions of skew Clifford algebras and provide several examples.
