The cone of positive semi-definite (SDP) matrices can be described by an infinite number of linear constraints. It is well-known that one can optimize over such a feasible area by standard Cutting Planes, but work on this idea remains a rare sight, likely due to its limited practical appeal compared to Interior Point Methods (IPMs). We extend the standard Cutting Planes by upgrading the widely-used separation sub-problem to the following projection sub-problem: given an SDP matrix $X\succeq \zeros$ and a direction $D\in\R^{n\times n}$, find the maximum step length $t$ such that $X+tD \succeq \zeros$, i.e., such that $X+tD$ stays in the SDP cone. Exploiting this sub-problem, we use a projective Cutting Planes variant that possesses a capability not inherent to standard Cutting Planes: generate feasible solutions along the iterations. In fact, the proposed SDP algorithm can actually construct both a primal-feasible and a dual-feasible SDP solution at each iteration, which is a feature that does not exist (built-in) in standard IPMs. The main theoretical challenge was to design a very fast algorithm to calculate this non-orthogonal projection in the SDP cone. Compared to other methods, Projective Cutting-Planes performs best on easily-feasible SDP programs with large (and dense) $n\times n$ matrices (e.g., $n\geq 1000$), because in such cases many IPMs seem to become computationally slower and memory-intensive. An interesting new feature is the natural integration of re-optimization: in contrast to IPMs, Projective Cutting-Planes does not need to restart from scratch when a new linear constraint is added after solving an initial SDP program. This enabled us to apply Branch-and-bound on a binary SDP program in a streamlined manner: we construct the branching tree incrementally by solving a sequence of slightly different SDP programs, since each branching rule reduces to imposing a specific constraint on the relaxed continuous SDP program.