I noticed that Entry of the Gladiators is said to be in C major scale but I was wondering if it could be in the chromatic scale since it uses all the notes. And if not, are there examples of songs that are considered to be in the chromatic scale?
Colloquially, we don't say pieces are "in the chromatic scale," no. We can say that a piece is in C major, or even just in C (not specifying major or minor), but not that something is in the chromatic scale.
One reason this might be so is due to the inherent hierarchy of tonality. If a piece is in C (like your example), arguably the two most important pitches will be C and G. A♯, however, will be comparatively low on that hierarchy. Therefore, saying a piece is in the key of "chromatic" or even "C chromatic" starts to muddy up the hierarchy that gave the piece sense of tonic in the first place. In contrast, saying a piece is "in C" gives us an immediate sense of that tonal hierarchy.
Around approximately 1920, composers such as Arnold Schoenberg viewed the increasing chromaticism of the earlier century (by composers like Wagner, Bruckner, and Mahler) as moving inexorably towards complete chromaticism. This is ultimately what spurred the notion of atonal music, which is music without a tonic. One branch of atonal music is serial music, wherein a piece is constructed using a pre-determined matrix of pitches. (This is a very basic definition, but it works for this answer.)
I say this because, in music without a tonic, we could in theory speak of it as being "in the chromatic scale" since there is no hierarchy of pitches. (Schoenberg famously spoke of the "democracy of tones" in this music.) But still it boils down to the fact that we just don't say something is "in the chromatic scale"; we'd just say it's atonal (or non-tonal) instead.
"Entry of the Gladiators" is in F in the partitura I have (the instruments in B look like it's in G, of course).
At any rate, it is totally not a piece in "the chromatic scale" since its harmonies are very straightforward progressions for the F major scale. The melody line may look largely chromatic but it is organized in a manner where the beats get in-harmony notes for the straightforward F major progression. If you look closely, you'll find that the chromatic lines constituting the melody line have very well-planned inflection points and sometimes have whole notes there in order to reach target on time.
This continues in the second part (where the bassoons carry the melody): here the first phrase in the bassoon is chromatic but its conclusion, which needs to cover a wider range, is diatonic.
The movements after that are more majestic, meaning that while there still is some chromaticity in the lead, notes are more or less all on-beat which requires harmonies to follow along more. But it's nevertheless mostly straightforward diatonic focus on the harmonies, starting with the Trio in B♭ major.
Given that in a standard major key, 7 out of the 12 available notes are pretty well bound to feature in any piece, there are only 5 other notes left and these can be found as accidentals in many pieces of music, often as passing notes rather than something which signifies a modulation or key change.
Unless we're talking atonal pieces, as indicated in some answers, a piece will have a specific key - just like Entry of the Gladiators. The fact that the other 5 non-diatonic notes are found in it as well has no bearing on a 'chromatic scale', even though some parts of it actually feature that same chromatic run as played in a chromatic scale.
Even if there was a 'chromatic key' - and there isn't- what key signature might it have, and why? As in my comment, 'scale' and 'key' are related, but not interchangeable.