Sr$_2$CuWO$_6$ is a double perovskite proposed to be at the border between two and three dimensional magnetism, with a square lattice of $S=frac{1}{2}$ Cu$^{2+}$ ions. We have used inelastic neutron scattering to investigate the spin wave excitations of the system, to find out how they evolve as a function of temperature, as well as to obtain information about the magnetic exchange interactions. We observed well defined dispersive spin wave modes at $6$~K, which partially survive above the magnetic ordering temperature, $T_N=24$~K. Linear spin wave theory is used to determine the exchange interactions revealing them to be highly two-dimensional in nature. Density functional theory calculations are presented supporting this experimental finding, which is in contrast to a previous emph{ab-initio} study of the magnetic interactions. Our analysis confirms that not the nearest neighbour, but the next nearest neighbour interactions in the tetragonal $ab$ plane are the strongest. Low incident energy measurements reveal the opening of a $0.6(1)$~meV gap below $T_N$, which suggests the presence of a very weak single ion anisotropy term in the form of an easy axis along $hat{mathbf{a}}$.
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