One has $\mathrm{H}^1_{\mathrm{et}}(X,\mathbf{G}_m) = \mathrm{H}^2(X,\mathbf{Z}(1)) = \mathrm{Pic}(X)$ (étale and motivic cohomology).

Is étale or motivic cohomology in other dimensions also representable by a scheme?

For the Brauer group $\mathrm{H}^2_{\mathrm{et}}(X,\mathbf{G}_m) = \mathrm{H}^3(X,\mathbf{Z}(1)) = \mathrm{Br}(X)$, this is false: https://mathoverflow.net/questions/185630/why-is-there-no-brauer-scheme

But what about other dimensions and other coefficients than $\mathbf{G}_m$? For example, one has for $f: X \to S$ and $\ell$-adic cohomology $\mathrm{R}^1f_*\mathbf{Z}_\ell(1) = T_\ell\mathbf{Pic}_{X/S}$.

Also, the **formal Brauer group** is pro-representable: http://www.math.uni-bonn.de/people/huybrech/K3Global.pdf p. 389, Chapter 18, Corollary 1.13.

(I assume that $X/S$ is "nice".)