  The Helmholtz decomposition splits a sufficiently smooth vector field into a
gradient field and a divergence-free rotation field. Existing decomposition
methods impose constraints on the behavior of vector fields at infinity and
require solving convolution integrals over the entire coordinate space. To
allow a Helmholtz decomposition in $\mathbb{R}^n$, we replace the vector
potential in $\mathbb{R}^3$ by the rotation potential, an n-dimensional,
antisymmetric matrix-valued map describing $n(n-1)/2$ rotations within the
coordinate planes. We provide three methods to derive the Helmholtz
decomposition: (1) a numerical method for fields decaying at infinity by using
an $n$-dimensional convolution integral, (2) closed-form solutions using
line-integrals for several unboundedly growing fields including periodic and
exponential functions, multivariate polynomials and their linear combinations,
(3) an existence proof for all analytic vector fields. Examples include the
Lorenz and R\"{o}ssler attractor and the competitive Lotka-Volterra equations
with $n$ species.
