Taylor series expansion up to third-order in a vector form

Question

I want to expand any function $f(x) \in \mathbb{R}^{n_f}$, $x\in \mathbb{R}^{n_x}$ near $\hat{x} \in \mathbb{R}^{n_x} $ with the help of Taylor series up to third-order in a vector form. Taylor series expansion up to second-order can be given as \begin{equation} f(x)=f(\hat{x})+f_x(x-\hat{x})+\sum_{i=1}^{n_f} e_i (x-\hat{x})^T f_{xx}^i(x-\hat{x}), \end{equation} where $e_i$ is $n_f$-dimensional Cartesian basis vector (having $i^{th}$ term unity and the rest elements zero), $f_x$ is ${n_f\times n_x}$ Jacobian matrix, defined as \begin{equation*} f_x=[\nabla_x f(x)^T]^T|_{x=\hat{x}}=\frac{\partial f}{\partial x}, \end{equation*} and Hessian of the $i$-th component of $f$ ($f^i_{xx}$ is ${n_x \times n_x}$ matrix)
\begin{equation*} f^i_{xx}=[\nabla_x \nabla_x^T f^i (x)]|_{x=\hat{x}} =\frac{\partial^2 f}{\partial x^2}. \end{equation*}