Is there a general expression for the moments (mean and variance) of the distribution of a maximum likelihood estimator?

I know that for large sample sizes that the distribution of the mle estimator will be Gaussian with mean given by the true value of the parameter, and variance given by the Fisher information.

However, for small sample sizes will the moments still be the same. That is, if I had sample sizes of only 3, is it correct to take the mean of the mle estimator as the true value of the parameter, and variance at the Fisher information?