I understand that semidefinite programming (SDP) is a subset of conic programming (CP). According to Boyd's Convex Optimization book section 4.6.1, the conic problem in inequality form is written as
$$ \begin{array}{ll} \text{minimize} & c^Tx \\ \text{subject to} & Fx + g \preceq_K 0 \end{array} $$
and in the next section, 4.6.2, a semidefinite program in inequality form is written as
$$ \begin{array}{ll} \text{minimize} & c^Tx \\ \text{subject to} & x_1 A_1 + \cdots + x_n A_n \preceq B \end{array} $$
So if we have set up a problem as a semidefinite program with known $A_i,B$ how can we rewrite it in the conic form?
This wiki page gives an idea of how the SDP is represented by vectors like in the CP. However, it is not clear to me how you rewrite the SDP in the CP form.