Consider the following classical construction (which is called Pfaffian representation as Sasha indicates):


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 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*.



Real Pfaffian representations of  real cubic surfaces
In other words you are asking about Pfaffian representations of a cubic hypersurface. To find a Pfaffian representation of a smooth cubic hypersurface of dimension $d$ is equivalent to constructing a vector bundle $E$ of rank 2 with $c\_1 = 2$ generated by $6$ global sections and with $H^\bullet(S,E(-k)) = 0$ for $1 \le k \le d$. So, you are asking whether the moduli space of such bundles on a cubic surface is nonempty and connected. 


As far as I know the moduli space is nonempty even for 3-dimensional cubics (and a fortiori for cubic surfaces). Also I know that in dimension 3 the moduli space is connected (in fact it is birational to the intermediate Jacobian of the cubic 3-fold).


