The Fisher information matrix (FIM) plays a key role in statistics. It is crucial in practice not only for evaluating the precision of parameter estimations, but also for statistical testing or experimental designs. In many latent variable models, the evaluation of the observed FIM is nonelementary due to the unobserved variables. Several methods have then been proposed to estimate the observed FIM. Among the most frequently used approaches are Monte-Carlo methods or iterative algorithms derived from the missing information principle. These methods however require to compute second derivatives of the complete data log-likelihood which has some disadvantages from a computational point of view. We propose an estimator of the observed FIM which only requires to compute the first derivatives of the complete data log-likelihood with respect to the parameters. Indeed, we consider the empirical first order moment estimate of the covariance matrix of the score. We derive a stochastic approximation algorithm for computing this estimator when its expression is not explicit. We study the asymptotic properties of the proposed estimator and of the corresponding stochastic approximation algorithm. Some numerical experiments are performed in mixed-effects models and mixture models to illustrate the theoretical results.