Development of Clifford algebra is based on the insights of Grassmann, Hamilton, and Clifford from the 19th century, yet at present, they still are an active area of mathematical research. The main objective of the paper is to exhibit a construction of a matrix algebra isomorphic to a Clifford algebra of signature (p,q), which can be automatically implemented using general-purpose linear algebra software. While this is not the most economical way of implementation for lower-dimensional algebras it offers a transparent mechanism of translation between a Clifford algebra and its faithful real matrix representation. Examples of lower-dimensional Clifford algebras are presented.