Is there a simple way to describe all $O(n)$ algorithms given simple assumptions about the machine?

Question

For example, can all $O(n)$ algorithms (where $n$ is strictly an integer) be described as:

for k in 0..f(n):
   O(1)(k)

where $f$ is a linear polynomial in $\Bbb{Z}[n]$ is included in the description,

and where $O(1)(k)$ above simply means some $O(1)$ operation that might reference the variable $k$. ?

Thanks.