Magnetic force between two charged particles?

I know the gravitational force between two particles with masses $m_1$, $m_2$:

$$ \vec{F} = \frac{Gm_1m_2}{|\vec{r}|^2} \hat{r} $$

And I know the electrical force between two particles with charges $q_1$, $q_2$:

$$ \vec{F} = -\frac{Kq_1q_2}{|\vec{r}|^2} \hat{r} $$

($\vec{r}$ is the position vector of particle 2 from the referential of particle 1 and $\hat{r} = \frac{\vec{r}}{|\vec{r}|}$)

I've been looking for an expression like these for the magnetic force since 2012... Then I found Physics Stack Exchange. Does anybody know if such formula exists?