In this paper, we propose a new method for the identification of linear Multiple Inputs-Multiple Outputs (MIMO) systems. By introducing a particular user-defined matrix that does not change the rank of the extended observability matrix when multiplying this latter matrix on the left, the subspace identification problem is recasted into a simple least squares problem with all regressors available. Therefore, the Singular Value Decomposition algorithm which is a customary tool in subspace identification can be avoided, thus making our method appealing for recursive implementation. The technique is such that the state coordinates basis of the estimated matrices is completely determined by the aforementioned user-defined matrix, that is, given such a matrix, the state basis of the identified matrices does not change with respect to the realization of input-output data.