We consider a Bose gas in spatial dimension $n\geq3$ with a repulsive, radially symmetric two-body potential $V$. In the limit of low density $\rho$, the ground state energy per particle in the thermodynamic limit is shown to be $(n-2)|\mathbb S^{n-1}|a^{n-2}\rho$, where $|\mathbb S^{n-1}|$ denotes the surface measure of the unit sphere in $\mathbb{R}^n$, and $a$ is the scattering length of $V$. Furthermore, for smooth and compactly supported two-body potentials, we derive an upper bound to the ground state energy with a correction term $(1+\gamma)8\pi^4a^6\rho^2|\ln(a^4\rho)|$ in $4$ dimensions, where $0<\gamma\leq C\|V\|_{\infty}^{1/2}\|V\|_1^{1/2}$, and a correction term which is $\mathcal{O}(\rho^2)$ in higher dimensions. Finally, we use a grand canonical construction to give a simplified proof of the second order upper bound to the Lee-Huang-Yang formula, a result first obtained by Yau and Yin. We also test this method in $4$ dimensions, but with a negative outcome.