Proving a general formula for the boost transformation of the electromagnetic field

In inertial frame $mathcal{O}$, a region of space-time is filled with constant electric field $vec{E}$ and magnetic field $vec{B}$. Another inertial frame $mathcal{O}’$ has 3-veclocity $vec{V}$ relative to $mathcal{O}$. What is the electromagnetic field $left(vec{E}’, vec{B}’right)$ measured in $mathcal{O}’$? Express the result in terms of $vec{E}$, $vec{B}$, $vec{V}$, dot product ($cdot$) and cross product ($times$).

This is to get the general formula for the boost transformation of the electromagnetic fields. I know the general form of the Lorentz boost transformation. So, obviously the solution for this problem seems to be applying this boost transformation to the electromagnetic field tensor $F^{uv}$. That is, for the boost transformation $Lambda^u_v$, calculate $F’^{ab}=Lambda^a_u Lambda^b_vF^{uv}$. But this seems like a tremendous amount of work… Is there any more efficient solution than this? Could anyone suggest me?