How to find the matrix representation of a given many-qubit Hamiltonian?

I have the following Hamiltonian

H = - Z1Z2 - Z2Z3 - Z1Z3 - 6(Z1 + Z2 + Z3)

Here, Z1, Z2, Z3 represent the Pauli-Z operators acting on qubits 1, 2, and 3, respectively. The interaction terms Z1Z2, Z2Z3, and Z1Z3 indicate that there is a ferromagnetic coupling between adjacent qubits. How can I simplify this further by substituting for Z1Z2, Z2Z3, Z1Z3 etc. in tensor product form in computational basis? Something like this (not sure how this can be obtained)

Z1Z2 = (Z ⊗ Z) (|1⟩⟨1| ⊗ |0⟩⟨0| + |0⟩⟨0| ⊗ |1⟩⟨1|)