It's probably better to see if any validated indexes/tests exist, and some do e.g. https://doi.org/10.1177%2F0013164492052002016
Despite its relevance to a wide variety of situations, competitiveness is a personality characteristic that has not been widely studied. Although some research in need for achievement, sports psychology, experimental social psychology, and personality assessment has approached the topic of competitiveness, few studies have provided a clear definition of the construct or a psychometrically-sound way of measuring it. This paper clarifies the conceptual definition of competitiveness and introduces a 20-item scale called the Competitiveness Index. Study 1 focuses on the validity and reliability of the measure; Study 2 reports the results of an exploratory factor analysis of the Competitiveness Index which yielded three factors: (a) Emotion, (b) Argument, and (c) Games.
But it's 1990's paper with only ~100 citations or so, so not a blockbuster. You should also look at which papers cite it, to see if anything better has been developed.
Also I found a follow-up/companion paper by same reserch group, published nearly at the same time as the previous paper (this one has fewer citations) https://doi.org/10.1016/0191-8869(92)90030-S
Little research has been devoted to the study of competitiveness as a personality trait. The Competitiveness Index (CI), developed by Smither and Houston (in press, Educational and Psychological Measurement), is a 20-item structured personality instrument that responds to this problem. This study explored the validity of the CI by investigating both the internal and external validity of the measure. The CI was completed by 255 subjects (158 nurses and 97 attorneys). A confirmatory factor analysis was conducted to assess the stability of the CI's factor structure across different population samples. In addition, a logistic discriminant analysis was performed to evaluate the CI's ability to differentiate between individuals in a competitive occupation (attorney) and a less competitive occupation (staff nurse). The results provide support for both the internal and external validity of the CI. The findings from the confirmatory factor analysis indicate that the factor structure of the CI is consistent across different population samples. The results of the discriminant analysis demonstrate that the CI can differentiate between individuals in more and less competitive occupations. Future research directions and possible applications of the CI are discussed.
And if need something briefer, a much more recent paper (2012) citing the previously mentioned work did this:
Competitiveness. To measure competitiveness, we modified the competitiveness index
scale suggested by Houston and colleagues (see Houston, Farese, & La Du, 1992; Smither &
Houston, 1992) so that we could capture state competitive mindset. The scale included the
following items: 1) Right now, I think that competing against an opponent would be enjoyable
and 2) Right now, I think that keeping score is important when playing games. The two items
were correlated at r=.73, p<.001, which corresponds to $\alpha$=.84 (M=4.95, SD=1.42).
