Consider the following classical construction:

Let $ V^4\subset \Lambda^2 \mathbb R^6$ be a four-dimensional subspace of the space of alternating two-forms. Then the equation $a\wedge a\wedge a=0$, $a\in V^4$ is homogeneous of degree three and hence defines a cubic surface in ${\mathbb P}V^4$.

Question. Can every real cubic surface be obtained by the above construction? If yes, is the set of corresponding representations for each cubic non-empty and connected? I would be grateful for a reference if there is one.

I am primarily interested in real case but if you only can comment on the complex case, this would be interesting for me as well.

As Sasha says, such cubics are called Pfaffian cubics.