In an earlier paper, the present authors (2013) introduced the altermatic number of graphs and used Tucker's Lemma, an equivalent combinatorial version of the Borsuk-Ulam Theorem, to show that the altermatic number is a lower bound for the chromatic number. A matching graph has the set of all matchings of a specified size of a graph as vertex set and two vertices are adjacent if the corresponding matchings are edge-disjoint. It is known that the Kneser graphs, the Schrijver graphs, and the permutation graphs can be represented by matching graphs. In this paper, as a generalization of the well-known result of Schrijver about the chromatic number of Schrijver graphs, we determine the chromatic number of a large family of matching graphs by specifying their altermatic number. In particular, we determine the chromatic number of these matching graphs in terms of the generalized Turan number of matchings.