Two electrons in a quantum dot (QD) can form triplet states. We analyze the exchange coupling of the triplet states of the negatively charged exciton in a QD (X$^-$, two electrons and one hole) with the spin of a magnetic atom (Mn). Two techniques are used to access the spin structure of this magnetic complex: the resonant excitation of the excited states of X$^-$-Mn and the analysis of the emission of a negatively charged biexciton in a magnetic dot (XX$^-$-Mn). The photo-luminescence (PL) excitation of X$^-$-Mn reveals excited states with a circularly polarized fine structure which strongly depends on the Mn spin state S$_z$ and gives rise to negative circular polarization emission. This fine structure arises from the coupling of the triplet states of an excited charged exciton with the Mn (X$^{-*}$-Mn) and its S$_z$ dependence can be described by a spin effective model. The recombination of XX$^-$-Mn leaves in the dot a charged exciton in its excited state and the PL structure is controlled by the coupling of triplet states of X$^{-*}$ with the Mn spin. An analysis of the polarization and magneto-optic properties of this emission gives access to the electron-hole exchange interaction within the triplets states. Comparing the fine structure of the singlet X$^-$-Mn and of the triplets of X$^{-*}$-Mn we can independently study the different source of anisotropy in the QD: the valence band mixing and the exchange interaction in an anisotropic potential
