First of all, we can project four-dimensional objects onto three dimensions, and then further project those onto 3 dimensions. See
https://en.wikipedia.org/wiki/Tesseract
The picture shown in this article is itself a 2D image, but you can surely make a 3D version of it.
Second, there is a pretty standard way of graphically representing arbitrary $n$-dimensional real vectors — using a bar chart. The discrete $x$ axis in this case corresponds to coordinate indices, while the continuous $y$ axis represents the values of corresponding components. This would be a picture for a vector in $\mathbb{R}^6$:
You may argue that this representation, unlike drawing an arrow, is very sensitive to basis changes, but this would not be fair, as we are simply more used to arrow rotations.
Importantly, the suggested representation is incredibly powerful in the sense that it immediately allows one to trace the connection between finite- and infinite-dimensional vector spaces. Indeed, if we agree to put all the $n$ components of a vector on the interval $\overline{AB}$ of a fixed length along the $x$ axis, it will become clear how in the limit $n\to\infty$ a finite-dimensional vector turns into a continuous function. In this case, $x$ simply becomes a continuous index! Furthermore, this illustration naturally explains the definition of scalar product of two real functions, $\langle f|g\rangle=\int_{A}^B f(x) g(x)\operatorname{d}x$ it is no more than the generalized component-wise product.
