Eisenstein series identities based on partial fraction decomposition

From the theory of modular forms, there are exactly $[(k-2)\mathrm{/}6]$ linear relations among the Eisenstein series $E_k$ and its products $E_{2i}E_{k-2i}
(2\le i \le [k/4])$. We present explicit formulas among these modular forms based on the partial fraction decomposition, and use them to determining a basis of the space of modular forms of weight $k$ on ${\mathrm{S}\mathrm{L}}_2(\mathbb{Z})$.