We introduce an estimation method for the scaled skewness coefficient of the sample mean of short and long memory linear processes. This method can be extended to estimate higher moments such as curtosis coefficient of the sample mean. Also a general result on computing all asymptotic moments of partial sums is obtained, allowing in particular a much easier derivation of some existing central limit theorems for linear processes. The introduced skewness estimator provides a tool to empirically examine the error of the central limit theorem for long and short memory linear processes. We also show that, for both short and long memory linear processes, the skewness coefficient of the sample mean converges to zero at the same rate as in the i.i.d. case.