  To an arc $\mathcal{A}$ of $\mathrm{PG}(k-1,q)$ of size $q+k-1-t$ we
associate a tensor in $\langle \nu_{k,t}(\mathcal{A})\rangle^{\otimes k-1}$,
where $\nu_{k,t}$ denotes the Veronese map of degree $t$ defined on
$\mathrm{PG}(k-1,q)$. As a corollary we prove that for each arc $\mathcal{A}$
in $\mathrm{PG}(k-1,q)$ of size $q+k-1-t$, which is not contained in a
hypersurface of degree $t$, there exists a polynomial $F(Y_1,\ldots,Y_{k-1})$
(in $k(k-1)$ variables) where $Y_j=(X_{j1},\ldots,X_{jk})$, which is
homogeneous of degree $t$ in each of the $k$-tuples of variables $Y_j$, which
upon evaluation at any $(k-2)$-subset $S$ of the arc $\mathcal{A}$ gives a form
of degree $t$ on $\mathrm{PG}(k-1,q)$ whose zero locus is the tangent
hypersurface of $\mathcal{A}$ at $S$, i.e. the union of the tangent hyperplanes
of $\mathcal{A}$ at $S$. This generalises the equivalent result for planar arcs
($k=3$), proven in \cite{BaLa2018}, to arcs in projective spaces of arbitrary
dimension. A slightly weaker result is obtained for arcs in
$\mathrm{PG}(k-1,q)$ of size $q+k-1-t$ which are contained in a hypersurface of
degree $t$. We also include a new proof of the Segre-Blokhuis-Bruen-Thas
hypersurface associated to an arc of hyperplanes in $\mathrm{PG}(k-1,q)$.
