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 [2], 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 $${\mathrm {PG}}(k-1,q)$$.