For a canonical transformation that is not a point transformation:
In the symplectic formulation of Hamiltonian mechanics we have
$$
\dot{\boldsymbol{\eta}} = J \frac{\partial H}{\partial \boldsymbol{\eta}},
$$
where $H$ is the Hamiltonian of the system,$\boldsymbol{\eta} = (q_1 \dots q_n p_1 \dots p_n)^T$ and the $2n \times 2n$ matrix J is defined by
$$
J = \begin{bmatrix}
0 & I \\
-I & 0
\end{bmatrix},
$$
with $I$ denoting the $n\times n$ identity matrix, and $0$ the $n\times n$ matrix with only $0$ as entries. Consider a coordinate transformation $\boldsymbol{\xi} = \boldsymbol{\xi}(\boldsymbol{\eta})$. Then
$$
\dot{\xi}_i = \frac{\partial \xi_i}{\partial \eta_j} \dot{\eta}_j = \frac{\partial \xi_i}{\partial \eta_j} J_{jk} \frac{\partial H}{\partial \eta_k} = \frac{\partial \xi_i}{\partial \eta_j} J_{jk} \frac{\partial \xi_l}{\partial \eta_k} \frac{\partial H}{\partial \xi_l}.
$$
That is,
$$
\dot{\boldsymbol{\xi}} = D J D^T \frac{\partial H}{\partial \boldsymbol{\xi}},
$$
where $D$ is the Jacobian matrix of the transformation. For the coordinate transformation to be a canonical transformation we thus require $DJD^T = J$. It is straight forward to show that this is equivalent to
$$
\lbrace \xi_i, \xi_j \rbrace_{\boldsymbol{\eta}} = J_{ij},
$$
which the preservation of Poisson brackets.
Now consider the determinant of $J$:
$$
\det(J) = \det(DJD^T) = \det(J) (\det{(D)})^2 \Rightarrow \vert \det(D) \vert = 1.
$$
This is condition is the same as conservation of the volume of a phase space volume element:
$$
\text{d}\omega ^\prime = \prod_i \text{d}q_i \text{d}p_i = \prod_i | \det(D) | \text{d}Q_i \text{d}P_i = \text{d}\omega
$$
