We introduce a family of SO(n)-symmetric spin chains which generalize the transverse-field Ising chain for n = 1. These spin chains are defined with Gamma matrices and can be exactly solved by mapping to n species of itinerant Majorana fermions coupled to a static Z2 gauge field. Their phase diagrams include a critical point described by the SO(n)1 Wess-Zumino-Witten model as well as two distinct gapped phases. We show that one of the gapped phases is a trivial phase and the other realizes a symmetry-protected topological phase when n ≥ 2. These two gapped phases are proved to be related to each other by a Kramers-Wannier duality. Furthermore, other elegant structures in the transverse-field Ising chain, such as the infinite-dimensional Onsager algebra, also carry over to our models.