High School Electric Field Created by a Finite Plate

[Heisenberg7]

Let's assume that we have a finite plate which is at the center of a cartesian coordinate system. Now let's define a point $r$ with coordinates $(0, 0, z)$. My question is, can we use Gauss's law to find the electric field at this point? The direction of the electric field is going to be up so all the $x$ and $y$ components will cancel out leaving us with only $z$ components of all the electric fields created by the infinitesimal points on the plane. Now, would it be possible for us to create a gaussian surface like a cylinder and calculate the electric field at the point $r$?

I've seen a video about this and the guy explaining it uses some pretty advanced calculus to find the electric field at the point r. In the end, he gets this equation: $$\vec{E}(\vec{r}) = \frac {Q}{4\pi\epsilon_oab} \arctan(\frac {ab}{2z \sqrt{a^2+b^2+4z^2}})$$

 

[Heisenberg7]

I guess this is a dumb question because there is basically little to no symmetry (only along the z axis is the electric field pointed upwards).

 

[Orodruin]

You are already onto it, but this may help:
https://www.physicsforums.com/insights/a-physics-misconception-with-gauss-law/