Teaching determinants poses significant challenges to the instructor of a proof-based undergraduate linear algebra course. The standard definition by cofactor expansion is ugly, lacks symmetry, and is hard for students to use in proofs. We introduce a visual definition of the determinant that interprets permutations as arrangements of non-attacking rooks on an "n × n" chessboard. We show that under this definition, many of the usual lemmas about determinants admit natural, insightful proofs that students themselves can readily discover.