Let $\mathbf{u}:\mathbb{R}^3\to\mathbb{R}^3$ be a smooth enough vector field that satisfies the following equation
$$\nabla\mathbf{u}+\nabla\mathbf{u}^T=\mathbf{C},\tag{1}$$
where $\nabla\mathbf{u}$ is gradient of $\mathbf{u}$ and $^T$ denotes transpose of a second order tensor and $\mathbf{C}$ is a constant symmetric second order tensor.
What is a neat way to solve $(1)$? I mean I am not really inclined to write down the components and solve each scalar equation. I want a basis-free approach. The final answer is
$$\mathbf{u}(\mathbf{x})=\mathbf{c}_1+\mathbf{c}_2\times\mathbf{x}+\mathbf{C}\cdot\mathbf{x}\tag{2}$$
where $\mathbf{c}_1$ and $\mathbf{c}_2$ are two constant vectors, $\cdot$ is simple contraction (or generalized dot product) and $\times$ denotes the usual cross product.