Can I describe whole Electromagnetism only with electric field $vec{E}$?

Is it possible to write all Maxwell’s equations only in terms of $vec{E}$ (without $vec{B}$)? The common formulation of Electromagnetism is governed by the equations below:

The Lorentz Force: $$vec{F} = q( vec{E} + vec{v} times vec{B}) $$

Maxwell’s equations:
$$vec{nabla} cdot vec{E} = frac{rho}{epsilon_0}$$
$$vec{nabla} times vec{E} = – frac{partial vec{B}}{partial t}$$
$$vec{nabla} cdot vec{B} = 0$$
$$c^2 vec{nabla} times vec{B} = frac{vec{j}}{epsilon _0} + frac{partial vec{E}}{partial t}$$

I have heard that $vec{B}$ is only a mathematical creation of beauty, and can be written as $$vec{B} = – frac{vec{r}}{r} times frac{vec{E}}{c},$$ where $vec{r}$ is distance vector from the charge to the point at which we want to detect the field. Couldn’t the equations be represented as below?

"Lorentz Force":
$$vec{F} = qvec{E} (1 – frac{dot{r}}{r}) – q (frac{vec{v}}{c}cdotvec{E})frac{vec{r}}{r}$$

"Maxwell’s Equations" (only using $vec{E}$ field):

$$vec{nabla} cdot vec{E} = frac{rho}{epsilon_0}$$
$$vec{nabla} times vec{E} = frac{partial}{partial t} (frac{vec{r}}{r} times frac{vec{E}}{c})$$
$$vec{nabla} times (frac{vec{r}}{r} times vec{E}) = – frac{vec{j}}{epsilon_0 c}-frac{1}{c} frac{partial vec{E}}{partial t}$$

Perhaps not all the equations above are strictly needed, but the question I have is whether or not I could write all of Maxwell’s Equations without $vec{B}$.