Consider the family of Wishart distributions on the space $V$ of real symmetric $p\times p$ matrices $\boldsymbol x=((x_{ij}))$. Then dimension of $V$ is $\frac12p(p+1)$ and a matrix $\boldsymbol x$ in this case can be considered as a vector $(x_{ij})_{1\le i\le j\le p}$. This is the 'vectorization' of $\boldsymbol x$.
If $V^+$ is the set of positive definite matrices of $V$, the Wishart distribution $W_p(n,\Sigma)$ (with $n\ge p$) has density function
$$f(\boldsymbol x\mid \Sigma)=\frac1{C_p(n)|\Sigma|^{n/2}}\exp\left\{-\frac12\operatorname{tr}(\Sigma^{-1}\boldsymbol x)\right\}|\boldsymbol x|^{(n-p-1)/2}\mathbf1_{V^{+}}(\boldsymbol x)\,, \tag{1}$$
where $C_p(n)$ is the normalizing constant and $\Sigma \in V^+$.
If you reparameterize $\boldsymbol\theta=-\frac12\Sigma^{-1}$, then keeping in mind the vectorization of $\boldsymbol\theta$,
$$-\frac12\operatorname{tr}(\Sigma^{-1}\boldsymbol x)=\sum_{1\le i\le j\le p}\theta_{ij}x_{ij}$$
And $(1)$ is now equivalent to
$$f(\boldsymbol x\mid \boldsymbol\theta)=h(\boldsymbol x)\exp\left\{\boldsymbol\theta^T \boldsymbol x - q(\boldsymbol\theta)\right\} \tag{2}\,,$$
with $$h(\boldsymbol x)=\frac1{C_p(n)}|\boldsymbol x|^{(n-p-1)/2}\mathbf1_{V^{+}}(\boldsymbol x) \quad, \quad q(\boldsymbol\theta)=\frac{n}2 \ln|\Sigma|=-\frac{n}2 \ln|-2\boldsymbol\theta|$$
So for fixed $n(\ge p)$, we can say that $f(\cdot\mid \Sigma)$ is a member of a natural exponential family with natural parameter $\boldsymbol\theta=-\frac12\Sigma^{-1}$ and sufficient statistic $T(\boldsymbol x)=\boldsymbol x$.
