Here we define decomposable pseudometrics. A pseudometric is decomposable if it can be represented as the sum of two pseudometrics that are obtained in a way other than the multiplication all distances by a positive factor. We consider spaces consisting of $n$ points. We prove that there exist a finite number of indecomposable pseudometrics (that is, a basis) such that any pseudometric is a linear combination of basic pseudometrics with nonnegative coefficients. For $n\le7$, the basic pseudometrics are listed. A decomposability test is derived for finite pseudometric spaces. We also establish some other conditions of decomposability and indecomposability.