Symmetric tensors are multi-indexed arrays whose entries are invariant with respect to permutations of multi-indices. Generating polynomials are linear relations of recursive patterns about tensor entries. A set of all generating polynomials can be represented by a matrix, which is called a generating matrix. Generally, a symmetric tensor decomposition can be uniquely determined by a generating matrix. We characterize the sets of such generating matrices and investigate their properties (e.g., the existence, dimensions, nondefectiveness). Using these properties, we propose computational methods for symmetric tensor decompositions. Extensive examples are shown to demonstrate their efficiency.