It is well known that in the Clifford algebra the Pfaffian plays a role analogous to the determinant in the Grassmann algebra. In this article with the help of the Pfaffian an operator on the tensor algebra is defined which generates the Clifford algebra from the tensor algebra. It is proved that the property of associativity of the Clifford algebra leads to an expansion of the Pfaffian analogous to the Laplace expansion of a determinant in terms of complementary minors.