This answer shows the relativistic formulation of electromagnetism and is heavily based on this document by David Tong.
In the final paragraph I somewhat give an answer to your question and hope others can correct/expand in case it's wrong.
In order to construct any relativistic theory you need to satisfy the postulates of special relativity, one of which requires the laws of physics to be the same in all inertial (not undergoing acceleration) frames of reference, or in other words your equations need to be Lorentz covariant. You achieve that by constructing your equations out of 4-vectors (e.g. the position vector $x^{\mu}=(ct,x,y,z)^T$) and tensors of higher rank.
As you can see, a 4-vector has 4 components, while the electric and magnetic fields $\vec{E}$ and $\vec{B}$ contain 3 components each - you need to do something extra in order to construct a 4-vector. Luckily, you know that electric and magnetic fields must obey Maxwell's equations, and you can define a scalar ($\phi$) and a vector ($\vec{A}$) potentials such as:
\begin{equation}
\vec{E} = -\nabla\phi-\partial{\vec A}/ \partial{t}
\end{equation}
\begin{equation}
\vec{B} = -\nabla\times\vec{A}.
\end{equation}
You can play around with these, e.g. shift $\vec{A}$ and make a corresponding shift in $\phi$ without changing the values of $\vec{E}$ and $\vec{B}$ in the process. I am writing all this because you can now define a new 4-vector using these potentials, $A^\mu= (\phi/c,\vec{A})^T$, and cast your equations in a Lorentz covariant form using it. More specifically, you can define the electromagnetic tensor as:
\begin{equation}
F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu = \begin{pmatrix}
0 & E_x/c & E_y/c & Ez/c \\
-E_x/c & 0 & -B_z & B_y \\
-E_y/c & B_z & 0 & -B_x \\
-E_z/c & -B_y & B_x & 0\\
\end{pmatrix}.
\end{equation}
If you apply a Lorentz boost to this object in order to go into a different inertial frame of reference, e.g. along the $x$-direction with velocity $v$, you will find the results (5.17) and (5.18) of Tong's document - no change in the $E_x$ and $B_x$, and mixing in the $y$ and $z$ components, e.g. $E_y'=\gamma(E_y-vB_z)$ - what appears to be the electric field in the primed coordinate along the $y'$-axis is a mixture of the electric field along the $y$ axis of the other frame in addition to some part of the magnetic field component along $z$. You can also define the 4-current using the tensor - $\partial_\nu F^{\nu\mu} = J^\mu$, where $J^\mu = (c\rho,\vec{j})$, $\rho$ - charge density, $\vec{j}$ - current.
About the question for other forces - there are equivalent to $F_{\mu\nu}$ tensors for the other forces as well (with some complications due to the weak and strong interactions being non-abelian), with potentials equivalent to $A_\mu$ and currents obtainable from the equations of motion. However, I am not aware of definitions of classical fields as components of the tensor (probably because definitions of classical fields makes less sense due to the short ranges of the weak/strong interactions).
