Yes, you can make it, but you also need to use a superposition principle.

 1. You determine that Couloms's law is a boundary case of the relativistic force, which acts of the charge.
 2. Using Lotentz transformation for the force and for the radius-vector,


$$
\mathbf F = \mathbf F' + \gamma \mathbf u \frac{\mathbf F' \cdot \mathbf v'}{c^{2}} + \Gamma \mathbf u \frac{(\mathbf u \cdot \mathbf F')}{c^{2}},
$$

$$
\mathbf r' = \mathbf r + \Gamma \mathbf u \frac{(\mathbf u \cdot \mathbf r)}{c^{2}} - \gamma \mathbf u t = \mathbf r + \Gamma \mathbf u \frac{(\mathbf u \cdot \mathbf r)}{c^{2}} (t = 0),
$$
where u is the speed of inertial system, v is the charge speed,
you can assume, that relative to the other inertial system with relative speed u the force looks as
$$
\mathbf F = q\mathbf E + \frac{q}{c}[\mathbf v \times \mathbf B],
$$
where 
$$
\mathbf E = \frac{\gamma Q \mathbf r}{(r^{2} + \frac{\gamma^{2}}{c^{2}}(\mathbf r \cdot \mathbf u)^{2})^{\frac{3}{2}}}, \quad \mathbf B = \frac{1}{c}[\mathbf u \times \mathbf E].
$$
3. After that, using primary theoremes of vector analisys and regularization procedure, you can "take" rot and div of the E and B expressions above. After that you can earn Maxwell's equations.