Many contributions have been made to the problem of characterizing the normal distribution using the property of independence of sample mean and sample variance, maximum likelihood, etc. In this paper, using certain identities among the product (linear) moments of order statistics in a random sample, the generalized truncated (both from below and above) normal distributions, the negative normal and the positive normal distributions are characterized in the class of arbitrary distributions having finite second moments. In Theorem 3.3, the normal distribution is characterized in the class of arbitrary distributions having mean zero and finite second moments. Bennett's [1] characterization of the normal distribution without the assumption of absolute continuity is a special case of Theorem 3.3, namely Corollary 3.3.2.